# Mathematical Food for Thought

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

• ## Meta

One By One, We’re Making It Fun. Topic: Calculus/S&S. May 28th, 2007

Theorem: (Stolz-Cesaro) Let and be sequences of real numbers such that is positive, strictly increasing, and unbounded. If the limit

exists, then the following limit also exists and we have

.

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Theorem: (Summation by Parts) If and are two sequences, then

.

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Problem: Let be a sequence of real numbers such that converges. Show that .

Solution: Define the sequence by and let . Then, by summation by parts with and , we have

.

The summation we wish to take the limit of is then

.

But since, by Stolz-Cesaro,

,

we obtain

.

QED.

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Comment: Summation by parts is a very useful technique to change sums around so that they are easier to evaluate. If you hadn’t noticed, it is the discrete analogue of integration by parts and is in fact very similar. Stolz-Cesaro is powerful as well and seems like a discrete analogue to L’Hopital (but I’m not sure about this one). Applying well-known calculus ideas to discrete things can often turn into neat results.

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Practice Problem: If is a decreasing sequence such that , show that

converges for all .