Question

what function is being represented by the following power series?

Answer 1

\[\sum_{0}^{\infty} \frac{ (-1)^{k}x ^{k+1} }{ 4^{k} }\]

Answer 2

Hint : geometric series

Answer 3

I see the numerator is similar to the sigma notation representation for sinx, cosx, and ln(1+x) but I cant

Answer 4

quite make out what series it should be.

Answer 5

\[ \sum_{0}^{\infty} \frac{ (-1)^{k}x ^{k+1} }{ 4^{k} } \\~\\
= 4\sum_{0}^{\infty} \frac{ (-1)^{k}x ^{k+1} }{ 4^{k+1} } \\~\\
= -4\sum_{0}^{\infty} \frac{ (-1)^{k+1}x ^{k+1} }{ 4^{k+1} } \\~\\
=-4\sum_{0}^{\infty} (-x/4) ^{k+1} \\~\\

\]

Answer 6

that is a geometric series with first term -x/4 and common ratio -x/4