# Understand the problem

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

##### Source of the problem

TIFR 2019 GS Part A, Problem 13

##### Topic

Analysis

##### Difficulty Level

Moderate

##### Suggested Book

Real analysis, Bartle & Sherbert

# Start with hints

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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