A calculus problem by Sabhrant Sachan

Calculus Level 4

$\displaystyle \lim_{n \to \infty} \left( \dfrac{1}{n+1}+\dfrac{1}{n+2}+\dfrac{1}{n+3}+\cdots+\dfrac{1}{n+10^{10^{100}}} \right) =\, ?$

Inspiration .

0

None of these choices

\ln{2}

\ln{3}

2 solutions

Shourya Pandey
Mar 4, 2017

We have

$0 \leq \displaystyle \sum_{i=1}^{10^{10^{100}}} \frac {1}{n+i} \leq \displaystyle \sum_{i=1}^{10^{10^{100}}} \frac {1}{n+1} = \frac {10^{10^{100}}}{n+1}$ , and by Sandwich theorem,

$0 \leq \displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^{10^{10^{100}}} \frac {1}{n+i} \leq \lim_{n \rightarrow \infty} \frac {10^{10^{100}}}{n+1} = 10^{10^{100}} \times \lim_{n \rightarrow \infty} \frac{1}{n+1} = 10^{10^{100}} \times 0 = 0,$

So $\displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^{10^{10^{100}}} \frac {1}{n+i} =0$

Nice solution. +1

Sabhrant Sachan - 4 years, 3 months ago

i still don't get it how can it be 0 it should be ln3 or may be little higher as after that it can be neglected , plz help

A Former Brilliant Member - 4 years, 2 months ago

Poonayu Sharma
Feb 28, 2017

When n tends to infinity, Question becomes 0+0+0+0....(10^10^100times) =0

Make sure you specify that you are adding finitely many zeros. Otherwise we might end up with something non-zero.

For example:

$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n+i} = \ln (2)$

due to Riemann Sums

Brandon Monsen - 4 years, 3 months ago

@Brandon Monsen yeah I'll edit in my solution. Thank you

Poonayu Sharma - 4 years, 3 months ago

A calculus problem by Sabhrant Sachan

2 solutions

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