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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 (sn)n≥1(s_{n})_{n}\ge_{1} and (tn)n≥1(t_{n})_{n}\ge_{1} defined by sn=∑k=0ns_{n} = \sum_{k=0}^{n} 1/(k!)21/(k!)^{2} and tn=∑k=0nt_{n} = \sum_{k=0}^{n} C(n, k)(−1)ᵏ/nᵏ. Which of the following statements are true?

  1. A.limsupn→∞limsup_{n\to\infty} tn≤t_{n} \le limsupn→∞limsup_{n\to\infty} sns_{n}✓ Correct
  2. B.limsupn→∞limsup_{n\to\infty} tn≤et_{n} \le e✓ Correct
  3. C.liminfn→∞liminf_{n\to\infty} sn≥e2s_{n} \ge e^{2}
  4. D.liminfn→∞liminf_{n\to\infty} tn≥et_{n} \ge 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.

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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

More on this topic

The chapter behind this: limsup and liminf — three equivalent definitions and the algebra — free to read

From The Real Line › limsup, liminf and subsequential limits

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