Find the sum, if possible

Question

2 + 5/2 + 25/8 + 125/32 + ...

Answer 1

Taking Andrew's process, we have an infinite sum.

 

∞

 ∑    2(1.25)^{(n - 1)}

n=1

 

where:

n=1  gives the first term in the sum

n=∞ gives the infinite term in the sum

 

Note: When infinity is the upper terminal in your summation sequence, the sum is infinite.  That is because the terms are increasing infinitely, thus the sum increases infinitely.

 

 

You will see something like this when you learn about convergence and divergence in calculus.

Answer 2

We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you won't get a final answer. The only possible answer would be infinity.

 

For this series...

The common ratio is 5/4 or 1.25

 

5/2 = 2(5/4)

25/8 = (5/2)(5/4)

125/32 = (25/8)(5/4)

and so forth

 

Thus, since the common ratio is >1 there is no actual sum for the infinite geometric series