Series Calculator

A series is the sum of the terms of a sequence. An infinite series takes the form:

$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$

The partial sums are $S_N = \sum_{n=1}^{N} a_n$. If the sequence of partial sums converges to a finite limit $S$, we say the series converges and $\sum_{n=1}^{\infty} a_n = S$. Otherwise, the series diverges.

Geometric Series: The series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1-r}$ when $|r| < 1$.

p-Series: The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges when $p > 1$ and diverges when $p \leq 1$.

Power Series: A series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n$ that represents a function within its radius of convergence.

Taylor Series: The power series expansion of $f(x)$ about $x = a$:

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

When $a = 0$, this is called a Maclaurin series.

Divergence Test (nth-term test)

If $\lim_{n \to \infty} a_n \neq 0$, the series diverges. Note: if the limit is 0, the test is inconclusive.

Ratio Test

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

• If $L < 1$: converges absolutely
• If $L > 1$: diverges
• If $L = 1$: inconclusive

Root Test

Compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. Same conclusion rules as the Ratio Test.

Integral Test

If $f(n) = a_n$ where $f$ is positive, continuous, and decreasing for $x \geq 1$:
$\sum_{n=1}^{\infty} a_n \text{ converges} \iff \int_1^{\infty} f(x)\,dx \text{ converges}$

Comparison Test

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

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

Alternating Series Test (Leibniz Test)

The alternating series $\sum (-1)^n b_n$ converges if:

1. $b_n > 0$ for all $n$
2. $b_n$ is decreasing
3. $\lim_{n \to \infty} b_n = 0$

Common Taylor/Maclaurin Series

Choosing the Right Test

• Misusing the Divergence Test: If $\lim a_n = 0$, this does NOT prove convergence. The harmonic series $\sum 1/n$ diverges even though $1/n \to 0$.
• Applying Ratio Test when L = 1: When the ratio limit equals 1, the test gives no information. You must use a different test.
• Confusing absolute and conditional convergence: A series can converge conditionally (like the alternating harmonic series) without converging absolutely.
• Wrong radius of convergence: Don't forget to check the endpoints separately when finding the interval of convergence.
• Taylor series remainder: The Taylor polynomial is only an approximation; for finite terms, there is a remainder term whose bound matters for accuracy.