How would I show that the series: 1 + 1/3 -1/2 +1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... converges to 3/2 ln 2? (Steps if you wouldn't mind would be helpful in my understanding)

Question

How would I show that the series: 1 + 1/3 -1/2 +1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... converges to 3/2 ln 2?  (Steps if you wouldn't mind would be helpful in my understanding)

amistre64 · Answer

find a rule for the series

amistre64 · Answer

might have to split it up into 3 sequences tho

anonymous · Answer

Are you sure that about
$$ \frac 3 2 \ln 2$$

amistre64 · Answer

$$\sum_{0}^{inf}\frac{1}{2n+1}-\sum_{0}^{inf} \frac{1}{2n+2}$$

anonymous · Answer

Yes I am confused also eliassaab , because I don't think  it converges to 3/2 ln 2

anonymous · Answer

@GavinxFiasco at what math class level this question was asked?

amistre64 · Answer

http://www.wolframalpha.com/input/?i=sum+%28%284n%2B3%29%282n%2B2%29%2B%284n%2B1%29%282n%2B2%29-%284n%2B1%29%284n%2B3%29%29%2F%28%284n%2B1%29%284n%2B3%29%282n%2B2%29%29+0+to+inf well, it is 3/2 ln(2)

anonymous · Answer

@eliassaab it was asked at first year university level.

anonymous · Answer

One can write the series as
$$ \sum_{k=0}^\infty \left(
\frac{1}{4 k+1}+\frac{1}{4
   k+3}-\frac{1}{2 k+2}\right) =
\sum_{k=0}^\infty \left(
\frac{8 k+5}{2 (k+1) (4 k+1)
   (4 k+3)}\right)
$$
And if you use Wolfram Alpha, you fine that the answer is $$ \frac 32 \ln 2$$

anonymous · Answer

oh thank you this makes alot of sense