Series

Infinite series

An infinite series (or just series) of constants is defined as a limit of finite series:

Geometric Series

A geometric series can be written in the form The sum of Geometric series is given by

new kind of series from CS370

Doesn’t have to sum to infinity where

  • Applying this for example to compute the number of nodes in a tree,

Telescoping Series

a telescoping series is a series whose general term can be written as the difference of two consecutive terms of a sequence, i.e.

Convergence / Divergence Tests

Convergence Test

Used on geometric series.

n^{th} Term Test (Test for Divergence)

If , then diverges.

DANGER: If , we cannot make any conclusions about divergence or convergence.

The Integral Test (Condition to use: all a_k > 0)

Theorem: Consider a series . Let be a function which is continuous, positive, and decreasing on , with for all .

  1. If converges, then converges.
  2. If diverges, then diverges.

Convergence of P-Series

Direct Comparison Test (Condition to use: a_k > 0)

Given a positive series . If we find another series , where

  1. , and if converges, then converges.
  2. , and if diverges, then diverges.

Limit Comparison Test ( a_k > 0)

Assume that the following limit exists (or ):

  • If (non-zero constant), then converges converges
    • They either both converge or both diverge
  • If , and converges, then also converges
  • If , and diverges, then also diverges

Alternating Series Test (AST) / Leibniz Test

Consider the series , where for every . If

  1. is a decreasing sequence, and
  2. ,

then the series converges.

Alternative Series Estimation Theorem (ASET)

Consider the series , where for every . If

  1. is a decreasing sequence, and
  2. ,

Then .

Assume that the following limit exists (or ):

  • If , then is absolutely convergent.
  • If , then is divergent.
  • If , then the test fails.

The Root Test

Assume that the following limit exists (or ):

  • If , then is absolutely convergent.
  • If , then is divergent.
  • If , then the test fails.

Conditional vs. Absolute Convergence

A series is called absolutely convergent if converges.

Absolute Convergence Implies Convergence

If the series converges, then also converges.

A series is said to be conditionally convergent if it converges, but the series diverges.

To check between for absolute convergence, we use The Ratio Test.