In order to uniquely define the geometric series, we need to know two things: the ratio between successive terms and at least one of the terms. Knowing one term doesn't give you the ratio of successive terms, but knowing two terms will give you the ratio. By the term generator for a geometric series , you can see that you only need two terms to find the ratio . 

The sum formula for  terms of an arithmetic series is .

For  terms, this formula becomes .

Using our term generator for  and , this formula becomes 

.

We can evaluate the infinite series by recognizing it as a geometric series times some constant.

Let's manipulate this series:

.

Now it suffices to evaluate , which we can recognize as the power series of  with , which is

 .

So we have

.

If a series is arithmetic, then there exists a common difference between each pair of consecutive terms in the series.

For this series

Because 

and

we find that there does NOT exist a common difference and as such,

the series is not arithmetic.

To determine the convergency of a geometric series, we must find the absolute value of the common ratio.

A geometric series will converge if the absolute value of the common ratio is less than one, or

In this problem, we see that 

And because 

we conclude that the series does not converge to a finite sum.

This is an arithmetic series.

Its general form is .

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

When k is not equal to 1, split the sums up into 2 piece and take the difference.

Find