Sequences and Series

Q: What is the difference between a sequence and a series?

A sequence is an ordered list of terms \{a_n\}. A series is the sum of those terms, written as \sum a_n, and it converges or diverges based on the behavior of its partial sums.

Q: What does it mean for an infinite series to converge?

An infinite series \sum_{n=1}^{\infty} a_n converges if its partial sums S_N=\sum_{n=1}^{N} a_n approach a finite limit as N\to\infty.

Q: Is \lim_{n\to\infty} a_n = 0 enough to guarantee convergence?

No. The condition \lim_{n\to\infty} a_n = 0 is necessary for convergence, but it is not sufficient. A series can still diverge even when its terms approach zero.

Q: What are the most common series types in calculus?

Common families include geometric series, p-series, alternating series, and power series. Recognizing the form helps you choose an appropriate convergence test.

Q: What is the difference between absolute and conditional convergence?

A series \sum a_n is absolutely convergent if \sum |a_n| converges. It is conditionally convergent if \sum a_n converges but \sum |a_n| diverges.

Q: What is a power series and what is the radius of convergence?

A power series has the form \sum_{n=0}^{\infty} c_n(x-a)^n. The radius of convergence R tells where the series converges around a: typically it converges for |x-a|<R and diverges for |x-a|>R, and the endpoints must be checked separately.

Sequences describe what happens to a list of terms as the index grows. Series track what happens to the running total when you add those terms. In calculus, the practical goal is almost always the same: decide convergence fast, recognize common patterns, and understand how power series lead into Taylor and Maclaurin approximations. The tools below cover the main convergence tests and power series topics, and they show the work step by step. Explore more topics in Calculus.

Test convergence, analyze series behavior, and work with Taylor and power series using structured tools.

Taylor-Maclaurin

Taylor Remainder Bound

Quick map of the workflow

Sequences: compute or estimate limits and use monotone or squeeze ideas when needed. Series: study partial sums and choose a convergence test that matches the structure of the terms. Power series: find the radius and interval of convergence, then check endpoints. Taylor and Maclaurin: build a polynomial approximation and use a remainder bound to control error.

Sequences: when they converge

A sequence $\{a_n\}$ converges to $L$ if $$\lim_{n\to\infty} a_n = L.$$ If the limit does not exist (or is infinite), the sequence diverges.

Geometric-type growth/decay: $r^n$ tends to $0$ when $|r|<1$ and blows up when $|r|>1$.
Power decay: $1/n^p \to 0$ for any $p>0$, but the speed of decay matters later for series.
Monotone + bounded: if $a_n$ is increasing and bounded above (or decreasing and bounded below), it converges.
Squeeze intuition: trap $a_n$ between two sequences that share the same limit.

Series: convergence via partial sums

A series $\sum a_n$ is judged by the behavior of its partial sums: $$S_N = a_1+a_2+\dots+a_N.$$ If $S_N$ approaches a finite number as $N\to\infty$, the series converges. If it doesn’t, it diverges. Convergence is about the running total, not about any single term on its own.

Convergence tests: how to choose

First, check the necessary condition: if $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 0$. If the term limit is not zero (or does not exist), the series diverges immediately.

Geometric series: looks like $ar^{n}$, converges when $|r|<1$.
$p$-series: $\sum \frac{1}{n^p}$ converges when $p>1$ and diverges when $p\le 1$.
Comparison test: compare $a_n$ to a known benchmark (often geometric or $p$-series).
Integral test: for positive decreasing terms tied to a nice $f(x)$, relate the series to an improper integral.
Ratio test: useful with factorials, exponentials, or products by comparing $a_{n+1}$ to $a_n$.
Root test: useful when $n$th powers appear by checking $n$th-root behavior.
Alternating series test: if signs alternate and $|a_n|$ decreases to $0$, the series converges.

Two outcomes show up constantly: absolute convergence (when $\sum |a_n|$ converges) and conditional convergence (when $\sum a_n$ converges but $\sum |a_n|$ diverges). Sign changes can help, but they do not replace the need to verify the conditions.

If you want to strengthen the limit intuition behind these ideas, continue to Limits and Continuity.

Power series and Taylor approximation

A power series has the form $$\sum_{n=0}^{\infty} c_n (x-a)^n.$$ Typically you find a radius $R$ so the series converges for $|x-a|<R$, diverges for $|x-a|>R$, and then you test the endpoints separately.

Taylor and Maclaurin series use derivatives to build a local polynomial approximation. The remainder term (often handled with a Taylor remainder bound such as the Lagrange form) measures the approximation error and tells you how accurate the polynomial is near the center. Taylor coefficients come from derivatives, so the most natural next topic is Derivatives.

Sequences and Series Calculator Categories

Work through limits, convergence, series types, and power series topics using focused calculators designed for clarity and step-by-step reasoning.

Keep exploring Mathematics or jump back to Calculators to browse more tools.

Questions About Sequences and Series

Answers related to limits, convergence, partial sums, power series, and Taylor approximations.

What is the difference between a sequence and a series? ⌄

What does it mean for an infinite series to converge? ⌄

Is $\lim_{n\to\infty} a_n = 0$ enough to guarantee convergence? ⌄

What are the most common series types in calculus? ⌄

What is the difference between absolute and conditional convergence? ⌄

What is a power series and what is the radius of convergence? ⌄

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