Suppose we have and we have . We have to find out the limit of the sequence.

TIFR 2019 GS Part A, Problem 13

Analysis

Moderate

Real analysis, Bartle & Sherbert

Do you really need a hint? Try it first!

I was observing that and and so on now it is clear that the function is increasing. Can you prove that the sequence is convergent and then how to find the limit?

If , then (the exponential is a growing function). You have a bounded increasing sequence, so it converges. Can you guess where it will converge?

Upon convergence, , or . The derivative of the LHS is , which has a single root, hence the function has at most two roots. By inspection, they are and . Can you get the answer now?

The iterations from do converge to .

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