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.
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 series using sigma notation, we need to establish the pattern of the terms.
Looking at the series, we notice that each term can be expressed as a power of a common ratio. Specifically, each term is the result of raising the common ratio (which is 5/8) to a power, starting from an initial term of 64.
To express this pattern using sigma notation, we start by defining the term number, which we'll call 'n'. The first term, 64, corresponds to n=0.
Now, to find the general term, we notice that the common ratio is (5/8), and each term is obtained by raising this ratio to the power of (n-1). Hence, the general term of the series is (64 * (5/8)^(n-1)).
Putting it all together, the series can be expressed in sigma notation as:
∑(64 * (5/8)^(n-1)), with n ranging from 1 to infinity.
Next, let's find the sum of this infinite series. Since this is a geometric series with a common ratio less than 1 in absolute value, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r),
where S is the sum, a is the first term, and r is the common ratio.
For our series, the first term a = 64 and r = 5/8.
Plugging these values into the formula, we get:
S = 64 / (1 - (5/8)).
Simplifying the expression, we have:
S = 64 / (3/8) = (64 * 8) / 3 = 512 / 3.
So, the sum of the infinite series is 512/3.
In summary, the series can be expressed using sigma notation as ∑(64 * (5/8)^(n-1)), and its sum is 512/3.