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Sum of x +1/x + x^2 + 1/x^2 + x^3 + 1/x^3 + infinity

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Sum of x +1/x + x^2 + 1/x^2 + x^3 + 1/x^3 + infinity  [#permalink]

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New post Updated on: 03 Feb 2019, 00:56
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A
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Question Stats:

29% (01:35) correct 71% (02:01) wrong based on 18 sessions

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Sum of \((x+\frac{1}{x})+(x^{2}+\frac{1}{x^{2}})+(x^{3}+\frac{1}{x^{3}})+.....\infty\) is


A) \(\frac{x+1}{-x}\)

B) \(\frac{x}{1+x}\)

C) \(\frac{x}{1-x}\)

D) \(0\)

E) \(-1\)

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Originally posted by 4d on 02 Feb 2019, 19:16.
Last edited by Bunuel on 03 Feb 2019, 00:56, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Sum of x +1/x + x^2 + 1/x^2 + x^3 + 1/x^3 + infinity  [#permalink]

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New post 02 Feb 2019, 19:44
1
1
The question can be rearranged as \(x+x^2+x^3+\)...... and \(\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\)......

The above expression is a GP series.
a = x
r = x
Sum of infinite series = \(\frac{a}{1-r}\) = \(\frac{x}{1-x}\)

For the second expression, a = 1/x, r = 1/x
Sum of infinite series = \(\frac{1}{x/(1-1/x)}\) = \(\frac{1}{x-1}\)

Adding both, \(\frac{x}{1-x} + \frac{1}{x-1}\) = \(\frac{x}{1-x} - \frac{1}{1-x}\) =\(\frac{x-1}{1-x}\) = -1.

E is the answer.
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Re: Sum of x +1/x + x^2 + 1/x^2 + x^3 + 1/x^3 + infinity  [#permalink]

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New post 02 Feb 2019, 19:50
4d wrote:
Sum of \((x+\frac{1}{x})+(x^{2}+\frac{1}{x^{2}})+(x^{3}+\frac{1}{x^{3}})+.....\infty\) is
A) \(\frac{x+1}{-x}\)
B) \(\frac{x}{1+x}\)
C) \(\frac{x}{1-x}\)
D) \(0\)
E) \(-1\)


Let us convert the equation in a bit of friendly terms...

\((x+\frac{1}{x})+(x^{2}+\frac{1}{x^{2}})+(x^{3}+\frac{1}{x^{3}})+.....\infty=(x+x^2+x^3+........)+(\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+.....)\)
So we have an infinite geometric progression and sum of such progressions is \(\frac{a}{1-r}\), where a is the first term and r is the ratio by which each term differes with the previos one.
\(=\frac{x}{1-x}+(1/x)/(1-(1/x))=\frac{x}{1-x}+\frac{1}{x-1}=\frac{x}{1-x}+\frac{1}{-(x-1)}=\frac{-(1-x)}{1-x}=-1\)

E
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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Re: Sum of x +1/x + x^2 + 1/x^2 + x^3 + 1/x^3 + infinity   [#permalink] 02 Feb 2019, 19:50
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