Calculus 2 - Series, Power Series & Convergence Tests Flashcards

Question 1

Geometric Series Test & Sum

Accepted Answer

A geometric series $\sum_{n=0}^{\infty} a r^n$ converges if $|r| < 1$ to the sum:
$$S = \frac{a}{1 - r}$$
If $|r| \ge 1$, the series **diverges**.

Question 2

$p$-Series Test

Accepted Answer

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$:
- **Converges** if $p > 1$
- **Diverges** if $p \le 1$

*Special case*: $p = 1$ is the Harmonic Series, which diverges.

Question 3

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

Accepted Answer

If $\lim_{n 	o \infty} a_n 
eq 0$ or the limit does not exist, then $\sum_{n=1}^{\infty} a_n$ **diverges**.

*Caution*: If $\lim_{n 	o \infty} a_n = 0$, the test is **inconclusive**.

Question 4

Integral Test

Accepted Answer

If $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$ with $f(n) = a_n$, then $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x) dx$ either **both converge** or **both diverge**.

Question 5

Direct Comparison Test (DCT)

Accepted Answer

Given $0 \le a_n \le b_n$ for all $n$:
1. If $\sum b_n$ converges, then $\sum a_n$ **converges**.
2. If $\sum a_n$ diverges, then $\sum b_n$ **diverges**.

Question 6

Limit Comparison Test (LCT)

Accepted Answer

For positive series $a_n, b_n > 0$, compute $L = \lim_{n 	o \infty} \frac{a_n}{b_n}$:
- If $0 < L < \infty$, both series **either both converge or both diverge**.
- If $L = 0$ and $\sum b_n$ converges, then $\sum a_n$ converges.
- If $L = \infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Question 7

Alternating Series Test (AST)

Accepted Answer

An alternating series $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ (where $b_n > 0$) **converges** if:
1. $b_{n+1} \le b_n$ for all $n$ (monotonically decreasing)
2. $\lim_{n 	o \infty} b_n = 0$

Question 8

Absolute vs. Conditional Convergence

Accepted Answer

- **Absolutely Convergent**: $\sum |a_n|$ converges (implies $\sum a_n$ converges).
- **Conditionally Convergent**: $\sum a_n$ converges, but $\sum |a_n|$ diverges.

*Example*: The alternating harmonic series $\sum \frac{(-1)^{n-1}}{n}$ is conditionally convergent.

Question 9

Ratio Test

Accepted Answer

Compute $L = \lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight|$:
- If $L < 1$, the series **converges absolutely**.
- If $L > 1$ (or $\infty$), the series **diverges**.
- If $L = 1$, the test is **inconclusive** (use another test).

Question 10

Root Test

Accepted Answer

Compute $L = \lim_{n 	o \infty} \sqrt[n]{|a_n|} = \lim_{n 	o \infty} |a_n|^{1/n}$:
- If $L < 1$, the series **converges absolutely**.
- If $L > 1$ (or $\infty$), the series **diverges**.
- If $L = 1$, the test is **inconclusive**.

Question 11

Alternating Series Estimation Theorem

Accepted Answer

If an alternating series satisfies the AST conditions, the remainder error $R_n = S - S_n$ when approximating sum $S$ by partial sum $S_n$ is bounded by the first neglected term:
$$|R_n| = |S - S_n| \le b_{n+1}$$

Question 12

Improper Integral of Type 1 (Infinite Interval)

Accepted Answer

An integral over an infinite domain defined as:
$$\int_{a}^{\infty} f(x) dx = \lim_{t 	o \infty} \int_{a}^{t} f(x) dx$$
It **converges** if the limit exists and is finite; otherwise, it **diverges**.

Question 13

Improper Integral of Type 2 (Infinite Discontinuity)

Accepted Answer

If $f(x)$ has an infinite discontinuity at $x = b$ on $[a, b]$:
$$\int_{a}^{b} f(x) dx = \lim_{t 	o b^-} \int_{a}^{t} f(x) dx$$
If the discontinuity is at $c \in (a, b)$, split into two integrals at $c$.

Question 14

Power Series & Radius of Convergence

Accepted Answer

A power series centered at $a$ has the form $\sum_{n=0}^{\infty} c_n (x - a)^n$.

The **Radius of Convergence** $R$ is found using the Ratio Test:
$$R = \lim_{n 	o \infty} \left| \frac{c_n}{c_{n+1}} ight|$$

Question 15

Finding the Interval of Convergence (IOC)

Accepted Answer

1. Use Ratio/Root Test to find the radius of convergence $R$.
2. Identify open interval $(a - R, a + R)$.
3. **Test Endpoints**: Separately test convergence at $x = a - R$ and $x = a + R$ using individual tests (AST, $p$-series, etc.) to include bracket `[` or parenthesis `)`.

Question 16

General Taylor & Maclaurin Series

Accepted Answer

The **Taylor series** of $f(x)$ centered at $x = a$ is:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$$
When centered at $a = 0$, it is called a **Maclaurin Series**:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$

Question 17

Maclaurin Series for $e^x$

Accepted Answer

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$
- **Radius of Convergence**: $R = \infty$
- **Interval of Convergence**: $(-\infty, \infty)$

Question 18

Maclaurin Series for $\sin(x)$

Accepted Answer

$$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$$
- Contains only **odd** powers.
- **Interval of Convergence**: $(-\infty, \infty)$, $R = \infty$

Question 19

Maclaurin Series for $\cos(x)$

Accepted Answer

$$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$$
- Contains only **even** powers.
- **Interval of Convergence**: $(-\infty, \infty)$, $R = \infty$

Question 20

Maclaurin Series for $\frac{1}{1-x}$ and $\ln(1+x)$

Accepted Answer

- $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots \quad 	ext{for } |x| < 1$
- $\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad 	ext{for } -1 < x \le 1$

Question 21

Taylor's Inequality (Remainder Bound)

Accepted Answer

If $|f^{(n+1)}(x)| \le M$ for $|x - a| \le d$, then the remainder $R_n(x) = f(x) - T_n(x)$ satisfies:
$$|R_n(x)| \le \frac{M}{(n+1)!} |x - a|^{n+1}$$

Calculus 2 - Series, Power Series & Convergence Tests

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