Question

Calculate the sum of $3^{-1},3^{-2},3^{-3},.....$

Answer 1

Number of terms $n=\infty$
Clearly, given series is G.P,
Common ratio $r=\frac{1}{3}$
First term $a=3^{-1}$
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$

If $n=\infty$ & $|r|<1$ then formula for total sum is
$S=\frac{a}{1-r}$
Substituting the values, we get
$S=\frac{3^{-1}}{1-3^{-1}}=\frac{1}{3-1}=\frac{1}{2}$