Series Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

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Use the divergence test on โˆ‘n=1โˆžn2n+1\sum_{n=1}^{\infty} \frac{n}{2n+1}.

Solution

  1. 1
    limโกnโ†’โˆžn2n+1=12โ‰ 0\lim_{n\to\infty} \frac{n}{2n+1} = \frac{1}{2} \neq 0.
  2. 2
    By the divergence test, the series diverges.

Answer

The series diverges.
If limโกanโ‰ 0\lim a_n \neq 0 the series must diverge. Here the terms approach 12\frac{1}{2}, not zero, so partial sums grow without bound.

About Series

The result of adding all the terms of a sequence together, either finitely or infinitely many terms.

Learn more about Series โ†’

More Series Examples

Example 1 easy

Compute partial sums [formula] through [formula] for [formula] and identify the limit.

Example 2 hard

Show that the harmonic series [formula] diverges.

Example 3 easy

Write the first four partial sums of [formula]

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