Question

Mr. Numbers, a mathematician, has been challenged by a rival mathematician to find the sum of a series that is infinite. If the series progresses as given, calculate its sum.

• A. $\frac{1}{2}$
• B. $1$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is C $\frac{2}{3}$

Taking "S" as the sum of the infinite series, we find:

$S=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}...$

$⟹\frac{S}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}...$

As there are an infinite number of terms in the series, removing one term from the series would still leave an infinite number of terms behind.

So, adding both the sums, we get:

$S+\frac{S}{2}=1+(-\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}-\frac{1}{4})+....$

$⟹\frac{3S}{2}=1$

$⟹S=\frac{2}{3}$