   Chapter 11.7, Problem 23E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.23. ∑ n = 1 ∞ tan ( 1 / n )

To determine

To test: Whether the series converges or diverges.

Explanation

Given:

The series is n=1tan(1n).

Result used:

(1) “Suppose that an and bn are the series with positive terms, if limnanbn=c, where c is a finite number and c>0, then either both series converge or both diverge.”

(2) The harmonic series n=11n is divergent.

Calculation:

Consider the series n=1bn=n=11n and n=1an=n=1tan(1n).

Obtain the limit of anbn.

limnanbn=limntan(1n)1n

Since limntan(1n)1n=00 is in an indeterminate form, apply L’Hospital’s Rule,

limnanbn=limn

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