1/2 + 3/4 + 5/8 + 7/16 + …

Putri Princezna
Tentukan hasil dari \dfrac{1}{2}+\dfrac{3}{4}+\dfrac{5}{8}+\dfrac{7}{16}+ \cdots

Hiyori Hinata

1+x+x^2+\cdots = \dfrac{1}{(1-x)} holds in (-1, 1). By using termwise differentiation, we get that

1+2x+3x^2+\cdots = \dfrac{1}{(1-x)^2}.

By substituting x with \frac{1}{2}, we have

\displaystyle\sum_{n=1}^{\infty}{\frac{n}{2^(n-1)}=4}

So

\begin{aligned}\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{5}{8}+\dfrac{7}{16}+ \cdots&=\displaystyle\sum_{n=1}^{\infty}{\frac{2n-1}{2^(n)}}\\&=\displaystyle \sum_{n=1}^{\infty}{\frac{n}{2^(n-1)}}-\sum_{n=1}^{\infty}{\frac{1}{2^(n)}}\\&=4-1= 3.\end{aligned}

Ashfaq Ahmad

\begin{aligned}S&= \frac{1}{2}+ \frac{3}{4}+ \frac{5}{8}+ \frac{7}{16}+\cdots\\ \frac{1}{2}S&= \frac{1}{4}+ \frac{3}{8}+ \frac{5}{16}+ \frac{7}{32}+\cdots\\ \text{So }&\\S-\frac{1}{2} S&= \frac{1}{2}+ \frac{2}{4}+ \frac{2}{8}+ \frac{2}{16}+\cdots\\ \frac{1}{2}S&= \frac{1}{2}+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots\\ \frac{1}{2}S&=\frac{1}{2}+1=\frac{3}{2}\\ \text{or }S&= 3\end{aligned}

 

Tulisan Terbaru :

Tentang msihabudin

Just a crazy people
Pos ini dipublikasikan di SOUL-MATE-MATIKA. Tandai permalink.

Tinggalkan Balasan

Isikan data di bawah atau klik salah satu ikon untuk log in:

Logo WordPress.com

You are commenting using your WordPress.com account. Logout / Ubah )

Gambar Twitter

You are commenting using your Twitter account. Logout / Ubah )

Foto Facebook

You are commenting using your Facebook account. Logout / Ubah )

Foto Google+

You are commenting using your Google+ account. Logout / Ubah )

Connecting to %s