Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Real Analysis Sequences and Series cheat sheet - grade college

Click image to open full size

Math Grade college

Real Analysis Sequences and Series Cheat Sheet

A printable reference covering sequence limits, convergence tests, Cauchy criteria, power series, and absolute convergence for college.

TopicsMathematicsFunctionsAlgebraExponentsGraphs
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Real analysis sequences and series study how ordered lists of numbers and infinite sums behave as their indices grow without bound. This cheat sheet helps students organize the main definitions, limit laws, and convergence tests used in proof-based calculus and analysis. It is especially useful for comparing similar tests and remembering the exact conditions needed before applying them.

The core ideas are convergence, divergence, boundedness, monotonicity, and Cauchy behavior. For series, the most important distinction is between convergence of the terms an→0a_n \to 0 and convergence of the infinite sum ∑n=1∞an\sum_{n=1}^{\infty} a_n. Power series add the ideas of radius and interval of convergence, usually found with the ratio or root test.

Many proofs depend on precise quantifiers, especially the ε\varepsilon definition of a limit and the Cauchy criterion.

Key Facts

  • A sequence (an)(a_n) converges to LL if for every ε>0\varepsilon > 0 there exists N∈NN \in \mathbb{N} such that n≥Nn \ge N implies ∣an−L∣<ε|a_n - L| < \varepsilon.
  • Every convergent sequence is bounded, but a bounded sequence does not have to converge.
  • A monotone increasing sequence that is bounded above converges, and a monotone decreasing sequence that is bounded below converges.
  • A series ∑n=1∞an\sum_{n=1}^{\infty} a_n converges if and only if its partial sums sn=∑k=1naks_n = \sum_{k=1}^{n} a_k converge.
  • If ∑n=1∞an\sum_{n=1}^{\infty} a_n converges, then an→0a_n \to 0, but an→0a_n \to 0 alone does not guarantee that ∑n=1∞an\sum_{n=1}^{\infty} a_n converges.
  • The geometric series satisfies ∑n=0∞arn=a1−r\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} when ∣r∣<1|r| < 1 and diverges when ∣r∣≥1|r| \ge 1.
  • The pp-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p} converges if p>1p > 1 and diverges if p≤1p \le 1.
  • For a power series ∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n(x-a)^n, the radius of convergence is often found from R=1lim sup⁡n→∞∣cn∣nR = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|c_n|}}.

Vocabulary

Sequence
A sequence is a function from the natural numbers to the real numbers, usually written as (an)(a_n).
Limit
The limit LL of a sequence is the number that the terms ana_n approach as n→∞n \to \infty.
Partial Sum
The nnth partial sum of a series is sn=∑k=1naks_n = \sum_{k=1}^{n} a_k, the finite sum of the first nn terms.
Absolute Convergence
A series ∑an\sum a_n converges absolutely if the positive series ∑∣an∣\sum |a_n| converges.
Conditional Convergence
A series ∑an\sum a_n converges conditionally if ∑an\sum a_n converges but ∑∣an∣\sum |a_n| diverges.
Radius of Convergence
The radius of convergence RR is the distance from the center aa within which a power series ∑cn(x−a)n\sum c_n(x-a)^n converges.

Common Mistakes to Avoid

  • Using an→0a_n \to 0 as a convergence test for ∑an\sum a_n is wrong because it is only a necessary condition, not a sufficient condition.
  • Applying the ratio test when the limit equals 11 is wrong because lim⁡n→∞∣an+1an∣=1\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = 1 gives no conclusion.
  • Forgetting endpoint checks in power series is wrong because the ratio or root test usually gives convergence only for ∣x−a∣<R|x-a| < R and divergence for ∣x−a∣>R|x-a| > R.
  • Assuming bounded means convergent is wrong because sequences such as an=(−1)na_n = (-1)^n are bounded but do not approach one limit.
  • Confusing absolute and conditional convergence is wrong because ∑∣an∣\sum |a_n| converging implies ∑an\sum a_n converges, but the reverse is not always true.

Practice Questions

  1. 1 Determine whether the sequence an=3n2−12n2+5a_n = \frac{3n^2 - 1}{2n^2 + 5} converges, and find its limit if it exists.
  2. 2 Test the series ∑n=1∞nn3+1\sum_{n=1}^{\infty} \frac{n}{n^3 + 1} for convergence or divergence.
  3. 3 Find the radius of convergence of the power series ∑n=1∞(x−2)nn3n\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}.
  4. 4 Explain why the convergence of ∑n=1∞∣an∣\sum_{n=1}^{\infty} |a_n| is a stronger condition than the convergence of ∑n=1∞an\sum_{n=1}^{\infty} a_n.

Understanding Real Analysis Sequences and Series

A useful way to think about a sequence is to ignore its early values. Limits describe what remains true far out in the list. This is why changing ten, one hundred, or any finite number of terms never changes a limit.

Proofs focus on controlling the tail of a sequence. The required starting index may depend on the chosen tolerance, which is the detail students most often miss. Subsequences are another powerful tool.

If two subsequences approach different values, the original sequence cannot settle to one value. For instance, a list that alternates between zero and one stays bounded but has two different subsequential behaviors.

For an infinite series, the important object is the running total. Each new term changes that total, so convergence means the changes eventually become too small to create a meaningful shift. The Cauchy viewpoint expresses this without naming the final total.

It asks whether every sufficiently far tail has a small total. This is especially useful when the sum is not known in advance. Cancellation needs careful attention.

A series with positive and negative terms can converge because opposite signs cancel. Absolute convergence is stronger because the series still converges after every term is replaced by its size.

This strength matters because it allows terms to be grouped or rearranged safely. A conditionally convergent series can change value under a rearrangement.

Convergence tests work by comparing a difficult series with a familiar pattern. The comparison test needs an inequality in the correct direction. To prove convergence of positive terms, students usually bound the unknown terms above by terms from a known convergent series.

To prove divergence, they bound them below by terms from a known divergent series. The ratio and root tests are most helpful when factorials, exponentials, or powers raised to the index appear. A result of one from either test gives no conclusion, so another method is needed.

It helps to check the term itself first. If the terms do not approach zero, no further test can rescue the series.

Power series behave like polynomials with infinitely many terms, but their behavior depends strongly on the input value. Inside the radius of convergence, the powers shrink quickly enough for the series to work well. Outside it, the powers grow too strongly.

At the boundary, each endpoint must be tested separately because the general ratio or root calculation usually says nothing there. Within its convergence interval, a power series can often be differentiated or integrated term by term. This makes it useful for approximating functions such as exponential, sine, and logarithm near a chosen center.

Students should track the center, radius, endpoint results, and whether convergence is absolute or conditional. These details determine which later operations are valid.