Sequences and Series

Sequences

Definition

A sequence is an ordered list of numbers $a_1, a_2, a_3, \ldots$ written $\{a_n\}_{n=1}^{\infty}$ or simply $\{a_n\}$. Each $a_n$ is called a term of the sequence. A series is the sum of the terms of a sequence: $\displaystyle\sum_{n=1}^{\infty} a_n$.

A sequence converges to a limit $L$ if $\displaystyle\lim_{n \to \infty} a_n = L$. Otherwise it diverges.

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Arithmetic Sequences

Definition

An arithmetic sequence has a constant common difference $d$ between consecutive terms:

$a_n = a_1 + (n - 1)d$

where $a_1$ is the first term and $d = a_{n+1} - a_n$ for all $n$.

Sum of an Arithmetic Sequence

The sum of the first $n$ terms is:

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\bigl[2a_1 + (n-1)d\bigr]$

Proof. Write the sum forward and backward:

$S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \cdots + a_n$ $S_n = a_n + (a_n - d) + (a_n - 2d) + \cdots + a_1$

Adding: $2S_n = n(a_1 + a_n)$, hence $S_n = \dfrac{n}{2}(a_1 + a_n)$.

Example. Find the sum of the first 50 positive odd numbers.

$a_1 = 1$, $d = 2$, $a_{50} = 1 + 49 \cdot 2 = 99$.

$S_{50} = \frac{50}{2}(1 + 99) = 2500$

Arithmetic Mean

The arithmetic mean of $a$ and $b$ is $\dfrac{a + b}{2}$. In an arithmetic sequence, each term is the arithmetic mean of its neighbours.

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Geometric Sequences

Definition

A geometric sequence has a constant common ratio $r$ between consecutive terms:

$a_n = a_1 \cdot r^{n-1}$

where $a_1$ is the first term and $r = \dfrac{a_{n+1}}{a_n}$ for all $n$.

Sum of a Finite Geometric Series

For $r \ne 1$:

$S_n = \frac{a_1(1 - r^n)}{1 - r} = \frac{a_1(r^n - 1)}{r - 1}$

Proof. $S_n = a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1}$.

$rS_n = a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1} + a_1 r^n$

Subtracting: $S_n - rS_n = a_1 - a_1 r^n$.

$S_n(1 - r) = a_1(1 - r^n) \implies S_n = \frac{a_1(1 - r^n)}{1 - r}$

Sum of an Infinite Geometric Series

If $|r| \lt 1$, the infinite geometric series converges:

$S_{\infty} = \sum_{n=1}^{\infty} a_1 r^{n-1} = \frac{a_1}{1 - r}$

If $|r| \ge 1$, the series diverges.

Example. Express $0.\overline{7}$ as a fraction.

$0.\overline{7} = \frac{7}{10} + \frac{7}{100} + \frac{7}{1000} + \cdots$

This is a geometric series with $a_1 = \dfrac{7}{10}$ and $r = \dfrac{1}{10}$.

$S_{\infty} = \frac{7/10}{1 - 1/10} = \frac{7/10}{9/10} = \frac{7}{9}$

Example. Evaluate $\displaystyle\sum_{n=0}^{\infty} \frac{3}{4^n}$.

$a_1 = 3$, $r = \dfrac{1}{4}$.

$S_{\infty} = \frac{3}{1 - 1/4} = \frac{3}{3/4} = 4$

Geometric Mean

The geometric mean of positive numbers $a$ and $b$ is $\sqrt{ab}$. In a geometric sequence with positive terms, each term is the geometric mean of its neighbours.

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Sigma Notation

Definition

$\sum_{k=m}^{n} a_k = a_m + a_{m+1} + \cdots + a_n$

Properties

$\sum_{k=m}^{n} c \cdot a_k = c\sum_{k=m}^{n} a_k$

$\sum_{k=m}^{n}(a_k + b_k) = \sum_{k=m}^{n} a_k + \sum_{k=m}^{n} b_k$

Useful Summation Formulas

Sum	Closed Form
$\displaystyle\sum_{k=1}^{n} k$	$\dfrac{n(n+1)}{2}$
$\displaystyle\sum_{k=1}^{n} k^2$	$\dfrac{n(n+1)(2n+1)}{6}$
$\displaystyle\sum_{k=1}^{n} k^3$	$\dfrac{n^2(n+1)^2}{4}$
$\displaystyle\sum_{k=0}^{n} r^k$	$\dfrac{1 - r^{n+1}}{1 - r}, \quad r \ne 1$

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Convergence Tests

The $n$-th Term Test (Divergence Test)

If $\displaystyle\lim_{n \to \infty} a_n \ne 0$, then $\displaystyle\sum a_n$ diverges.

Caution. If $\lim_{n \to \infty} a_n = 0$, the series may or may not converge. The harmonic series $\sum \dfrac{1}{n}$ diverges despite its terms tending to zero.

Comparison Test

If $0 \le a_n \le b_n$ for all $n$:

• If $\sum b_n$ converges, then $\sum a_n$ converges.
• If $\sum a_n$ diverges, then $\sum b_n$ diverges.

Ratio Test

For a series $\sum a_n$ with $a_n \ne 0$:

$L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$

• $L \lt 1$: the series converges absolutely.
• $L \gt 1$: the series diverges.
• $L = 1$: the test is inconclusive.

Example. Does $\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n}$ converge?

$L = \lim_{n \to \infty} \frac{(n+1)^2 / 2^{n+1}}{n^2 / 2^n} = \lim_{n \to \infty} \frac{(n+1)^2}{2n^2} = \frac{1}{2} \lt 1$

The series converges.

Integral Test

If $f$ is continuous, positive, and decreasing on $[1, \infty)$, then $\sum f(n)$ converges if and only if $\displaystyle\int_1^{\infty} f(x)\,dx$ converges.

Example. The harmonic series diverges because:

$\int_1^{\infty} \frac{1}{x}\,dx = \lim_{b \to \infty} \ln b = \infty$

Alternating Series Test (Leibniz Test)

An alternating series $\sum (-1)^{n+1} a_n$ (or $\sum (-1)^n a_n$) converges if:

1. $a_n$ is decreasing: $a_{n+1} \le a_n$ for all $n$.
2. $\displaystyle\lim_{n \to \infty} a_n = 0$.

Example. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges (the alternating harmonic series) since $\dfrac{1}{n}$ decreases to $0$.

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The Binomial Theorem

Finite Binomial Expansion

For $n \in \mathbb{'\{'}Z{'\}'}^+$:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

where the binomial coefficient is:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

General Binomial Expansion

For any $n \in \mathbb{'\{'}R{'\}'}$ and $|x| \lt 1$:

$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$

This is an infinite series that converges for $|x| \lt 1$ (and at the endpoints depending on $n$).

Example. Expand $(1 + x)^{-2}$ up to the term in $x^3$.

$(1 + x)^{-2} = 1 + (-2)x + \frac{(-2)(-3)}{2}x^2 + \frac{(-2)(-3)(-4)}{6}x^3 + \cdots$

$= 1 - 2x + 3x^2 - 4x^3 + \cdots$

Example. Find the coefficient of $x^3$ in the expansion of $(2 - 3x)^{1/2}$.

$(2 - 3x)^{1/2} = \sqrt{2}\left(1 - \frac{3x}{2}\right)^{1/2}$

$= \sqrt{2}\left[1 + \frac{1}{2}\!\left(-\frac{3x}{2}\right) + \frac{(1/2)(-1/2)}{2}\!\left(-\frac{3x}{2}\right)^{\!2} + \frac{(1/2)(-1/2)(-3/2)}{6}\!\left(-\frac{3x}{2}\right)^{\!3} + \cdots\right]$

The $x^3$ coefficient:

$\sqrt{2} \cdot \frac{-3/8}{6} \cdot \left(-\frac{27}{8}\right) = \sqrt{2} \cdot \frac{81}{384} = \frac{27\sqrt{2}}{128}$

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Maclaurin Series

Definition

The Maclaurin series of a function $f$ is its Taylor series expansion about $x = 0$:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$

Standard Maclaurin Series

Function	Maclaurin Series	Radius of Convergence
$e^x$	$\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$	$\infty$
$\sin x$	$\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$	$\infty$
$\cos x$	$\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$	$\infty$
$\ln(1 + x)$	$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$	$1$
$(1 + x)^n$	$\displaystyle\sum_{k=0}^{\infty} \binom{n}{k}x^k$	$1$
$\dfrac{1}{1 - x}$	$\displaystyle\sum_{n=0}^{\infty} x^n$	$1$
$\arctan x$	$\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}$	$1$

Deriving Series

Example. Find the Maclaurin series of $e^x$.

$f(0) = 1, \quad f'(0) = 1, \quad f''(0) = 1, \quad \ldots, \quad f^{(n)}(0) = 1$

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Example. Find the Maclaurin series of $\sin x$.

$f(0) = 0, \quad f'(0) = 1, \quad f''(0) = 0, \quad f'''(0) = -1, \quad f^{(4)}(0) = 0, \quad \ldots$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Maclaurin Series from Known Series

Example. Find the Maclaurin series of $x^2 e^x$.

Since $e^x = \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}$, multiplying by $x^2$:

$x^2 e^x = \sum_{n=0}^{\infty} \frac{x^{n+2}}{n!} = x^2 + x^3 + \frac{x^4}{2!} + \frac{x^5}{3!} + \cdots$

Example. Find the Maclaurin series of $\dfrac{1}{1 + x^2}$.

$\frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} (-1)^n x^{2n}$

Integrating term by term gives the series for $\arctan x$:

$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$

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Taylor Series

Definition

The Taylor series of $f$ about $x = a$ is:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$

This reduces to the Maclaurin series when $a = 0$.

Taylor Polynomial Approximation

The $n$-th degree Taylor polynomial of $f$ about $a$ is:

$T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k$

Lagrange Remainder

The error in approximating $f(x)$ by $T_n(x)$ is bounded by:

$|R_n(x)| = \left|f(x) - T_n(x)\right| \le \frac{M|x - a|^{n+1}}{(n+1)!}$

where $M$ is an upper bound for $|f^{(n+1)}(t)|$ for all $t$ between $a$ and $x$.

Example. Approximate $\sqrt{e}$ using a 3rd degree Maclaurin polynomial and bound the error.

$T_3(x) = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6}$.

$T_3(0.5) = 1 + 0.5 + 0.125 + 0.020833\ldots = 1.64583\ldots$

True value: $e^{0.5} \approx 1.64872$.

For the error bound, $|f^{(4)}(t)| = e^t \le e^{0.5}$ for $0 \le t \le 0.5$:

$|R_3(0.5)| \le \frac{e^{0.5} \cdot (0.5)^4}{24} \approx \frac{1.649 \cdot 0.0625}{24} \approx 0.0043$

Actual error: $|1.64872 - 1.64583| \approx 0.0029$, which is within the bound.

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Proof by Induction

Structure

Mathematical induction proves a statement $P(n)$ for all integers $n \ge n_0$:

1. Base case: Verify $P(n_0)$ is true.
2. Inductive hypothesis: Assume $P(k)$ is true for some arbitrary $k \ge n_0$.
3. Inductive step: Using the hypothesis, prove $P(k + 1)$ is true.
4. Conclusion: By the principle of mathematical induction, $P(n)$ is true for all $n \ge n_0$.

Summation Proofs

Example. Prove that $\displaystyle\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

Base case ($n = 1$): $\dfrac{1 \cdot 2 \cdot 3}{6} = 1$. True.

Inductive hypothesis: Assume $\displaystyle\sum_{k=1}^{j} k^2 = \frac{j(j+1)(2j+1)}{6}$ for some $j \ge 1$.

Inductive step:

$\sum_{k=1}^{j+1} k^2 = \sum_{k=1}^{j} k^2 + (j+1)^2 = \frac{j(j+1)(2j+1)}{6} + (j+1)^2$

$= \frac{(j+1)}{6}\bigl[j(2j+1) + 6(j+1)\bigr] = \frac{(j+1)}{6}\bigl[2j^2 + 7j + 6\bigr]$

$= \frac{(j+1)(j+2)(2j+3)}{6}$

This is the formula with $n = j + 1$. Hence $P(j+1)$ is true. By induction, the result holds for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

Divisibility Proofs

Example. Prove that $3^n - 1$ is divisible by $2$ for all $n \in \mathbb{'\{'}N{'\}'}$.

Base case ($n = 0$): $3^0 - 1 = 0$, which is divisible by $2$. True.

Inductive hypothesis: $3^k - 1 = 2m$ for some $m \in \mathbb{'\{'}Z{'\}'}$.

Inductive step:

$3^{k+1} - 1 = 3 \cdot 3^k - 1 = 3(2m + 1) - 1 = 6m + 3 - 1 = 6m + 2 = 2(3m + 1)$

This is divisible by $2$. By induction, the result holds.

Inequality Proofs

Example. Prove that $2^n \gt n$ for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

Base case ($n = 1$): $2^1 = 2 \gt 1$. True.

Inductive hypothesis: $2^k \gt k$ for some $k \ge 1$.

Inductive step:

$2^{k+1} = 2 \cdot 2^k \gt 2k \ge k + 1$

(The last inequality holds since $k \ge 1$.) Therefore $2^{k+1} \gt k + 1$. By induction, $2^n \gt n$ for all positive integers $n$.

warning

Common Pitfall

The inductive step must genuinely use the inductive hypothesis. Simply proving $P(k+1)$ independently of $P(k)$ is not a valid induction argument. Always make it explicit where the hypothesis is used.

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Additional Worked Examples

Worked Example: Convergence of a Series by Ratio Test

Determine whether $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{10^n}$ converges or diverges.

Solution

Apply the ratio test:

$L = \lim_{n \to \infty} \frac{(n+1)! / 10^{n+1}}{n! / 10^n} = \lim_{n \to \infty} \frac{(n+1)! \cdot 10^n}{n! \cdot 10^{n+1}} = \lim_{n \to \infty} \frac{n+1}{10} = \infty$

Since $L \gt 1$, the series diverges by the ratio test.

Worked Example: Maclaurin Series of a Composite Function

Find the Maclaurin series of $f(x) = e^{-x^2}$ up to the term in $x^6$, and use it to approximate $\displaystyle\int_0^{0.5} e^{-x^2}\,dx$.

Solution

Substitute $-x^2$ into the Maclaurin series for $e^u$:

$e^{-x^2} = 1 + (-x^2) + \frac{(-x^2)^2}{2!} + \frac{(-x^2)^3}{3!} + \cdots = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6} + \cdots$

Integrate term by term from $0$ to $0.5$:

$\int_0^{0.5} e^{-x^2}\,dx \approx \int_0^{0.5} \left(1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6}\right)dx$

$= \left[x - \frac{x^3}{3} + \frac{x^5}{10} - \frac{x^7}{42}\right]_0^{0.5}$

$= 0.5 - \frac{0.125}{3} + \frac{0.03125}{10} - \frac{0.0078125}{42}$

$= 0.5 - 0.041667 + 0.003125 - 0.000186 \approx 0.4613$

The actual value of the error function at $0.5$ gives approximately $0.4613$, confirming the accuracy of this approximation.

Worked Example: General Binomial Expansion to Find a Coefficient

Find the coefficient of $x^4$ in the expansion of $(1 - 2x)^{-3}$.

Solution

$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \frac{n(n-1)(n-2)(n-3)}{4!}x^4 + \cdots$

Here $n = -3$ and we substitute $x \to -2x$:

$(1 - 2x)^{-3} = 1 + (-3)(-2x) + \frac{(-3)(-4)}{2!}(-2x)^2 + \frac{(-3)(-4)(-5)}{3!}(-2x)^3 + \frac{(-3)(-4)(-5)(-6)}{4!}(-2x)^4 + \cdots$

The $x^4$ coefficient:

$\frac{(-3)(-4)(-5)(-6)}{24} \cdot (-2)^4 = \frac{360}{24} \cdot 16 = 15 \cdot 16 = 240$

So the coefficient of $x^4$ is $240$.

Worked Example: Induction Proof Involving Summation

Prove by induction that $\displaystyle\sum_{k=1}^{n} k \cdot 2^{k-1} = (n - 1) \cdot 2^n + 1$ for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

Solution

Base case ($n = 1$): LHS $= 1 \cdot 2^0 = 1$. RHS $= (1 - 1) \cdot 2^1 + 1 = 1$. True.

Inductive hypothesis: Assume $\displaystyle\sum_{k=1}^{j} k \cdot 2^{k-1} = (j - 1) \cdot 2^j + 1$ for some $j \ge 1$.

Inductive step:

$\sum_{k=1}^{j+1} k \cdot 2^{k-1} = \sum_{k=1}^{j} k \cdot 2^{k-1} + (j+1) \cdot 2^j$

$= (j - 1) \cdot 2^j + 1 + (j+1) \cdot 2^j \quad \mathrm{(by\ hypothesis)}$

$= \bigl[(j-1) + (j+1)\bigr] 2^j + 1 = 2j \cdot 2^j + 1 = j \cdot 2^{j+1} + 1$

$= (j + 1 - 1) \cdot 2^{j+1} + 1$

This is the formula with $n = j + 1$. By induction, the result holds for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

Worked Example: Taylor Series Error Bound

Use a second degree Taylor polynomial of $\ln(1 + x)$ about $x = 0$ to approximate $\ln(1.2)$. Bound the error.

Solution

$f(x) = \ln(1 + x), \quad f'(x) = \frac{1}{1+x}, \quad f''(x) = \frac{-1}{(1+x)^2}, \quad f'''(x) = \frac{2}{(1+x)^3}$

$T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 = 0 + x - \frac{x^2}{2}$

$T_2(0.2) = 0.2 - \frac{0.04}{2} = 0.2 - 0.02 = 0.18$

True value: $\ln(1.2) \approx 0.1823$.

For the error bound on $[0, 0.2]$: $|f'''(t)| = \dfrac{2}{(1+t)^3} \le 2$ for $0 \le t \le 0.2$.

$|R_2(0.2)| \le \frac{2 \cdot (0.2)^3}{6} = \frac{2 \cdot 0.008}{6} \approx 0.00267$

Actual error: $|0.1823 - 0.18| = 0.0023$, which is within the bound.

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Common Pitfalls

1. Misidentifying the first term in sigma notation. $\sum_{k=0}^{n}$ has $n + 1$ terms, while $\sum_{k=1}^{n}$ has $n$ terms. Confusing the starting index leads to off-by-one errors in sums.

2. Applying the infinite sum formula when $|r| \ge 1$. The formula $S_{\infty} = \dfrac{a_1}{1 - r}$ is valid only when $|r| \lt 1$. For $|r| \ge 1$ the series diverges and the formula is meaningless.

3. Computing the wrong term number. The $n$-th term of a geometric sequence is $a_1 r^{n-1}$, not $a_1 r^n$. Similarly, the $n$-th term of an arithmetic sequence is $a_1 + (n-1)d$, not $a_1 + nd$.

4. Using the ratio test when $L = 1$. The ratio test is inconclusive when $L = 1$. The series $\sum \dfrac{1}{n}$ diverges and $\sum \dfrac{1}{n^2}$ converges, yet both give $L = 1$.

5. Weak base case in induction. The base case must match the claim. If the statement starts at $n = 1$, proving it for $n = 0$ is not sufficient unless the domain is specified to include $0$.

6. Neglecting the alternating sign in the general binomial expansion. When $n$ is not a positive integer, the series is infinite and the sign of each coefficient depends on the value of $n$. Substituting $x \to ax$ also changes the sign of odd powers when $a \lt 0$.

7. Confusing the Lagrange remainder with the actual error. The remainder bound $|R_n(x)| \le \dfrac{M|x-a|^{n+1}}{(n+1)!}$ gives an upper bound, not the exact error. The actual error may be much smaller.

8. Forgetting convergence conditions for Maclaurin series. Each standard Maclaurin series has a specific radius of convergence. The series for $\ln(1+x)$ only converges for $-1 \lt x \le 1$; using it outside this interval gives incorrect results.

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Exam-Style Problems

1. Find the sum of the infinite geometric series $8 - 4 + 2 - 1 + \cdots$ and express the repeating decimal $0.\overline{27}$ as a fraction in lowest terms.

2. Use the ratio test to determine the convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{3^n}{n!}$.

3. Find the coefficient of $x^5$ in the expansion of $(1 + 3x)^{-2}$.

4. Prove by induction that $\displaystyle\sum_{k=1}^{n} \frac{1}{k(k+1)} = \frac{n}{n+1}$ for all $n \in \mathbb{'\{'}Z{'\}'}^+$.

5. Find the Maclaurin series of $\dfrac{x}{1 + x^2}$ up to $x^7$ and state the radius of convergence.

6. Use a third degree Maclaurin polynomial of $\cos x$ to approximate $\cos(0.3)$. Bound the error using the Lagrange remainder.

7. An arithmetic sequence has first term $5$ and common difference $3$. A geometric sequence has first term $2$ and common ratio $2$. Find the smallest $n$ for which the $n$-th term of the geometric sequence exceeds the $n$-th term of the arithmetic sequence.

8. Determine whether $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ converges absolutely, converges conditionally, or diverges.

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Cross-References

• Differentiation is needed to derive Maclaurin and Taylor series: see Differentiation
• Integration connects to term-by-term integration of power series: see Integration
• Differential equations use series expansions as solution methods: see Differential Equations
• Proof and reasoning techniques including induction: see Proof