Express the infinite series below using sigma notation and then find the sum.

64 + 40 + 25 + 15.625...

Any explanation would be excellent. Thank you so much.

To express the given infinite series using sigma notation, we need to understand the pattern in the terms.

Looking at the terms 64, 40, 25, 15.625, we can observe that each term is obtained by taking the previous term and dividing it by 2. In other words, each term is half of the previous one.

We can express this pattern in sigma notation as follows:

∑ (64 / 2^n),

where the summation is taken over the positive integers, n = 0, 1, 2, ...

Now, let's find the sum of this infinite series.

To find the sum, we need to calculate the sum of all the individual terms in the series.

Using the formula for the sum of a geometric series, we have:

Sum = a / (1 - r),

where a is the first term of the series and r is the common ratio.

In this case, the first term a is 64, and the common ratio r is 1/2.

Plugging these values into the formula, we get:

Sum = 64 / (1 - 1/2)

= 64 / (1/2)

= 64 * 2

= 128.

Therefore, the sum of the given infinite series is 128.