Question

is the sum of a series equal to what it converges to?
Find the sum of the convergent series.

Answer 1

\[\sum_{n=0}^{\infty}5/-2^n\]

Answer 2

the answer is got was -infinity but the real answer is -5/3

Answer 3

Do you mean
\[\sum_{n=0}^\infty\left(-\frac{5}{2}\right)^n\quad\text{or}\quad\sum_{n=0}^\infty5\left(-\frac{1}{2}\right)^n\quad\text{or something else altogether?}\]

Answer 4

\[\sum_{n=0}^{\infty}(\frac{ -5 }{ 2^{n}})\]

Answer 5

So the exponent doesn't apply to the -5, correct?

Answer 6

The exponent does not apply to -5.

Answer 7

Alright, so
\[\sum_{n=0}^\infty -\frac{5}{2^n}=-5\sum_{n=0}^\infty \left(\frac{1}{2}\right)^n\]
which is a geometric series with common ratio \(\dfrac{1}{2}\). Since the magnitude of the common ratio is less than 1, you know the series converges.

For a convergent geometric series of the form
\[\sum_{n=0}^\infty ar^n\]
the sum is
\[\frac{a}{1-r}\]