Reciprocal Sums

Calculus Level 3

Let $g = 10^{100}$ denote one googol. What is the value of

$\dfrac1g+\dfrac1{g+1}+\dfrac1{g+2} + \cdots?$

1

\frac1{ \sqrt{g} }

\frac1{10^{99} }

\frac1{g-1}

\infty

1 solution

William Whitehouse
Jun 28, 2016

Relevant wiki: Convergence - Comparison Test

First let me prove that the harmonic series diverges, that is 1+1/2 +1/3+1/4+...=Infinity.

We need to consider a different sequence: (1)+ (1/2)+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+... Now clearly every element in every bracket is less than or equal to a corresponding element in the harmonic series so this sequence is strictly less than the harmonic series. It is also clear that this sequence diverges as it can be simplified to an infinite number of 1/2 added together.

So we have shown that the harmonic series diverges the series in the question is the same sequence with the first (googol-1) terms removed. clearly the first (googol-1) terms sum to a finite number and an infinite number- a finite number is still infinite hence this sequence also diverges to infinity.

Reciprocal Sums

1 solution

0 pending reports