Question

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

Answer 1

Consider the following infinite series:

∞Σk=1-(-1/2)k-1

 

Check whether the above series is geometric or not.

The above series is an exponential function with a base of -1/2. So, the above series will be a geometric series with common ratio of r = -1/2.

Common ratio of the series will be -1 < (r = -1/2) < 1.

 

The common ratio of above geometric sequence is less than 1 and greater than -1. That is, the common ratio lies between -1 and 1.

 

Therefore, the sum of above infinite series is “Defined.”

 

Use the formula for the sum of an infinite geometric series,

S∞ = a1/(1 – r) ...... (1)

 

Substitute k = 1 in the explicit formula –(-1/2)k-1 and compute first term of above series,

a1 = -(-1/2)1-1

      = -(-1/2)0

     = -1

 

Substitute a1 = -1 and r = -1/2 in the formula (1) and simplify,

S∞ = (-1)/{1 – (-1/2)}

      = (-1)/(1 + 1/2)

     = (-1)/3/2

     = -2/3