Analysis Homework: Sequence Properties and the Stolz Theorem

My recent analysis homework has moved on from the previous sequence-limit exercises to the Stolz theorem. When I first looked at the proof in the classroom photos, I could not follow it. Why suddenly add the inequalities? Why divide by yₙ? These notes separate the definitions from the purpose of each proof step, so I have something concrete to check when reviewing.

Three different requirements on a sequence

For a real sequence aₙ, the definitions are:

• Convergence to a real number A: for every ε>0, there is a positive integer N such that every n>N satisfies |aₙ−A|<ε. All terms in a sufficiently late tail must stay close to the same number.
• Boundedness: there is a fixed M>0 such that every term satisfies |aₙ|≤M. One fixed range must contain the entire sequence.
• Divergence in absolute value: for every M>0, there is a positive integer N such that every n>N satisfies |aₙ|>M. All sufficiently late terms must remain outside any prescribed range.

A convergent sequence is bounded, but the converse fails: (−1)ⁿ is a counterexample. Divergence in absolute value implies unboundedness, but the converse fails too. Set the odd-indexed terms equal to n and the even-indexed terms equal to 0: the sequence is unbounded, yet it keeps returning to zero. The crucial words are “every sufficiently late term,” rather than just “some large terms.”

The third definition concerns absolute value. For (−1)ⁿn, the absolute values tend to positive infinity, but the sequence itself tends to neither positive nor negative infinity.

Check the denominator before using Stolz

The version studied here requires yₙ to be strictly increasing and yₙ→+∞. If the ratio of successive differences satisfies

(xₙ₊₁−xₙ)/(yₙ₊₁−yₙ)→L,

then xₙ/yₙ→L. Here L may be a finite real number or positive or negative infinity. There is no requirement that xₙ increase. The assumptions can be checked against Gabriel Nagy’s notes on the Stolz–Cesàro theorem.

The intuition is to pass from a ratio of increments to a ratio of accumulated quantities. Those two ratios need not be equal at a particular index. Strict increase makes yₘ₊₁−yₘ>0, so multiplying by it preserves inequalities. Growth to positive infinity makes the initial contribution disappear after division.

Why adding differences leaves only the endpoints

First take a finite L. Given ε>0, choose a starting index k with yₖ>0, beyond which every increment ratio lies between L−ε/2 and L+ε/2. Multiply by the positive denominator and add from m=k to n−1. The intermediate terms cancel:

(xₖ₊₁−xₖ)+…+(xₙ−xₙ₋₁)=xₙ−xₖ.

The same happens to the differences of y, giving

(L−ε/2)(yₙ−yₖ)<xₙ−xₖ<(L+ε/2)(yₙ−yₖ).

Divide by the positive number yₙ and rearrange to obtain the estimate

|xₙ/yₙ−L|<ε/2+|xₖ−Lyₖ|/yₙ.

The point I need to keep track of is that once ε is given, k is fixed; it is n that continues to grow. The second term on the right has a constant numerator and a denominator tending to positive infinity. Eventually it is less than ε/2, so the total error is less than ε. Adding and dividing each serve a specific purpose in the proof.

Checking the method with a sum of squares

The classroom example was (1²+2²+…+n²)/n³. Let the numerator be xₙ and the denominator be yₙ. After checking that n³ is strictly increasing and tends to positive infinity, calculate

xₙ₊₁−xₙ=(n+1)²,

yₙ₊₁−yₙ=3n²+3n+1.

The ratio of these differences tends to 1/3, so the original ratio also tends to 1/3 by Stolz. Taking a difference turns a long sum into its single newly added term.

The assumptions still matter. With xₙ=1 and yₙ=n/(n+1), the denominator increases but tends only to 1. The increment ratio is zero, while the original ratio tends to 1. This shows why growth of the denominator to infinity cannot simply be omitted.

These notes break the proof into steps; they do not mean I can already use it fluently. For the next review, the first task is to rewrite “multiply by the positive denominator—add—divide by yₙ—control the initial contribution” without the explanation in front of me, then check whether I can say where each assumption is used.