Part CCSIR NET June 2025the-sum-of-1-over-k-factorial-squared-is-not-the-square-of-the-sum

The sum of 1 over k factorial squared is not the square of the sum

Consider the sequences $(s_{n})_{n} \geq_{1}$ and $(t_{n})_{n} \geq_{1}$ defined by $s_{n} = \sum_{k = 0}^{n}$ $1/ (k!)^{2}$ and $t_{n} = \sum_{k = 0}^{n}$ C(n, k)(−1)ᵏ/nᵏ. Which of the following statements are true?

A. $l im s u p_{n \to \infty}$ $t_{n} \leq$ $l im s u p_{n \to \infty}$ $s_{n}$ ✓
B. $l im s u p_{n \to \infty}$ $t_{n} \leq e$ ✓
C. $l imin f_{n \to \infty}$ $s_{n} \geq e^{2}$
D. $l imin f_{n \to \infty}$ $t_{n} \geq e$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

The sum of 1 over k factorial squared is not the square of the sum

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