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Series expansion

  • Thread starter thercias
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  • #1
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Homework Statement


Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5.... (-1)^n (x^n)/(n+1) = ln(1+x)/x
with |x| < 1

Homework Equations




The Attempt at a Solution


finding derivative of the function multiplied by x
d/dx(xS(x))
= 1 -x + x^2 - x^3 + x^4 - x^5 +....

absolute value of this function = 1 + x + x^2 + x^3 + x^4.... = 1/1-x ( geometric series)
the integral of that gives us -ln(1-x) + c

drawing a blank now. not really sure where to go.
 

Answers and Replies

  • #2
haruspex
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= 1 -x + x^2 - x^3 + x^4 - x^5 +....

absolute value of this function = 1 + x + x^2 + x^3 + x^4.... = 1/1-x ( geometric series)
No it isn't. But it is a geometric series, so you can sum it quite easily.
 
  • #3
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ok the absolute value part is wrong, but what do you mean i can sum it up easily? i dont get it. is the integral still -ln(1-x) + c..? i know the geometric series for 1 + x + x^2 + .... = 1/1-x, but how does that change with alternating signs? is it going to be (-1)^n*(1/1-x)? so the integral is (-1)^n*-ln(1-x)? how do i go from that to ln(1+x)/x?
 
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  • #4
HallsofIvy
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An alternating series involves (1)n which can easily be incorported into the geometric series.
 
  • #5
haruspex
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An alternating series involves (1)n which can easily be incorported into the geometric series.
I think you meant (-1)n.
Thercias, what is the ratio of consecutive terms in your alternating series? If it's independent of n then it's a geometric series.
 

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