How do you test \sum _ { m = 1} ^ { \infty } \frac { ( - 6) ^ { m + 1} } { 4^ { 8m } } for convergence or divergence?

1 Answer
May 20, 2017

This series is convergent by the geometric series test.

Explanation:

sum_(m=1)^oo frac{(-6)^(m+1)}{4^(8m)}

Rewrite the fraction using exponent rules
sum_(m=1)^oo frac{(-6)^(m)(-6)}{32^(m)}

sum_(m=1)^(oo) (-6)((-6)/32)^m

sum_(m=1)^(oo) -6((-3)/16)^m

Because the absolute value of the common ratio |r| = |(-3)/16| = 3/16is less than 1, the series is convergent by the geometric series test.

To find what it converges to:
Geometric series that converge converge to a/(1-r) where a is the first term of the series, and r is the common ratio.

a/(1-r) = frac{-6(-3/16)}{1+ 3/16}

= ((9/8))/((19/16))

= 9/8 * 16/19

= 18/19