sum

Let $s = 1-\frac{1}{4}+\frac{1}{16}-\frac{1}{64}+\ldots$ .

$4s = 4-1+\frac{1}{4}-\frac{1}{16}+\ldots = 4-s$ .

Now we solve for $s$ :

$5s = 4 \longrightarrow s = \boxed{\frac{4}{5}}$ .

$1-\frac { 1 }{ 4 } +\frac { 1 }{ 16 } -\frac { 1 }{ 64 } +...\\ =\sum _{ t=0 }^{ \infty }{ \frac { 1 }{ (-{ 4) }^{ t } } } =\frac { 1 }{ 1-(-\frac { 1 }{ 4 } ) } \\ =\frac { 1 }{ \frac { 5 }{ 4 } } =\frac { 4 }{ 5 } =0.8$