Show that the sum of the series from j = 1 up to 2m of (-1)^(j+1)*(1/j) is equal to the sum of the series from j=1 up to m of (1/(m+j)).(adsbygoogle = window.adsbygoogle || []).push({});

it works for m = 1.

After assuming m = k is true, and plugging in m = k+1 i got

sum of the series from j = 1 up to 2(k+1) of (-1)^(j+1)*(1/j) is equal to the sum of the series from j=1 up to k+1 of (1/(k+j+1)).

Now the big problem im having is with the right side. The left side is ok, i realize that it can be written as the left side of the induction hypothesis plus the next two terms. Whats really bugging me tho is the right side. I cant see any way to form the induction hypothesis out of it. This is really because of the m+j in the series.

Because if i want to write the sum of the series from j=1 up to k+1 of (1/(k+j+1)), but with the upper limit as k, i would write it as:

the sum of the series from j=1 up to k of (1/(k+j+1)) + 1/2(k+1) (since j = k+1)

is that correct? And if it is i cant generate the induction left hand side. Meaning i cant get rid of the summation. Am i on the right track? where did i make a mistake? Any help would be much appreciated.

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# Another Proof by Induction

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