Convergence and Divergence Math Example 4

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

Example 4

medium
Apply the ratio test to ∑n=0∞3nn!\displaystyle\sum_{n=0}^{\infty} \frac{3^n}{n!}.

Solution

  1. 1
    an+1an=3n+1(n+1)!⋅n!3n=3n+1\frac{a_{n+1}}{a_n} = \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n} = \frac{3}{n+1}.
  2. 2
    L=lim⁡n→∞3n+1=0<1L = \lim_{n\to\infty} \frac{3}{n+1} = 0 < 1.
  3. 3
    The series converges (and the sum is e3e^3).

Answer

The series converges (sum =e3= e^3).
Factorial grows faster than any exponential. The ratio goes to 0, far below 1, confirming convergence. This is the Taylor series for exe^x evaluated at x=3x = 3.

About Convergence and Divergence

A series converges if the sequence of its partial sums approaches a finite limit. A series diverges if the partial sums grow without bound or oscillate without settling.

Learn more about Convergence and Divergence →

More Convergence and Divergence Examples

Example 1 medium

Use the ratio test to determine whether [formula] converges or diverges.

Example 2 hard

Determine whether [formula] converges using the [formula]-series test.

Example 3 easy

Does [formula] converge or diverge?

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