Diverging series limit

Calculus Level 4

{ a 0 = 1 a n + 1 = a n + e − a n for n = 0 , 1 , 2... \large \begin{cases} a_0 = 1 \\ a_{n+1} = a_n + e^{-a_n} & \text{for } n = 0, 1, 2 ... \end{cases}

Let a n a_n be defined as above, find lim ⁡ n → ∞ a n − log ⁡ ( n ) \displaystyle \lim_{n \to \infty} a_n - \log (n) .


The answer is 0.

Shivam Jadhav by Shivam Jadhav , 22, India

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2 solutions

Sundar R
Jun 29, 2017

Looking at the recursive formula, it is a discretization of the differential equation dy/dx = e^-x with solution : y=-e^(-x) + c and we need lim (x-> inf) y which equals 0

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Harsh Shrivastava
Jun 26, 2017

@Spandan Senapati . Can you post a solution, I solved by some approximations.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I could just guess it right away. I will try for a formal solution.

Spandan Senapati - 3 years, 11 months ago

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