Question

Find the sum of $\frac{1}{(1+x)(1+ax)}+\frac{a}{(1+ax)(1+a^{2}x)}+\frac{a^2}{(1+a^{2}x)(1+a^{3}x)}+....$ to $n$ terms.

Answer 1

The $n^{th}$ term$=\frac{a^{n-1}}{(1+a^{n-1}x)(1+a^{n}x)}=\frac{A}{1+a^{n-1}x}+\frac{B}{1+a^{n}x}$ suppose$;$
$\therefore a^{n-1}=A(1+a^{n}x)+B(1+a^{n-1}x)$.
Bu putting $1+a^{n-1}x,1+a^{n}x$ equal to zero in succession, we obtain
$A=\frac{a^{n-1}}{1-a},B=-\frac{a^n}{1-a}$.
Hence $u_1=\frac{1}{1-a}\left(\frac{1}{1+x}-\frac{a}{1+ax}\right),$
Similarly$,$ $u_2=\frac{1}{1-a}\left(\frac{a}{1+ax}-\frac{a^2}{1+a^{2}x}\right),$
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$u_n=\frac{1}{1-a}\left(\frac{a^{n-1}}{1+a^{n-1}x}-\frac{a^n}{1+a^{n}x}\right),$
$\therefore S_n=\frac{1}{1-a}\left(\frac{1}{1+x}-\frac{a^n}{1+a^{n}x}\right)$.