# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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To Converge Or Not To Converge. Topic: Real Analysis/S&S. November 27th, 2006

Problem: (Stanford Math 51H, Cauchy Root Test) Suppose there exists a and an integer such that for all . Prove that converges.

Solution: Well let’s just discard because it is finite and obviously converges. It remains to show that converges.

But then we have . So

.

The last summation, however, is a geometric series with common ratio , so it converges. Hence our sum does as well. QED.

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Comment: The root test is, in effect, a comparison to a geometric series. The hypothesis is that we can bound by a geometric series for all large , implying convergence.

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Practice Problem: (Stanford Math 51H) Discuss the convergence of

.

### 3 Responses to “To Converge Or Not To Converge. Topic: Real Analysis/S&S.”

1. t0rajir0u Says:

the inequality x ≥ sin x for positive x is quite easy to prove

2. Xuan Says:

“The word ‘easy’ should never be used in math.”
–gnay naux

3. blue_giraffe Says:

haha nice xuan =)