Identify the two series that are the same.
a. Sigma (lower n = 4; upper infinity) n(3/4)^n
b. Sigma (lower n = 0; upper infinity) (n+1)(3/4)^n
c. Sigma (lower n = 1; upper infinity) n(3/4)^(n-1)
To identify the two series that are the same, we need to simplify each series and compare the expressions for the general term.
Let's start with series a:
a. Sigma (lower n = 4; upper infinity) n(3/4)^n
The general term for this series is n(3/4)^n.
Now let's simplify series b:
b. Sigma (lower n = 0; upper infinity) (n+1)(3/4)^n
The general term for this series is (n+1)(3/4)^n.
To compare the two general terms, let's rewrite the second general term to make it more similar to the first one:
(n+1)(3/4)^n = n(3/4)^n + 1(3/4)^n
As you can see, the general term for series b can be broken down into two terms: n(3/4)^n from series a and 1(3/4)^n.
Since the two series have the same terms except for a constant term, we can deduce that series a and series b are the same.
Therefore, the two series that are the same are:
a. Sigma (lower n = 4; upper infinity) n(3/4)^n
b. Sigma (lower n = 0; upper infinity) (n+1)(3/4)^n