Answer

(a) We can rewrite the series so the terms are fractions whose denominators are powers of 2:

From here it makes sense to view the series as starting at *n* = 3 and having general term

Then we can write the series in sigma notation as

(b) Way 1: We already saw that we can rewrite the series as

In order to think of the first term as using *n* = 2 instead of *n* = 3, we can break up the denominators like this:

Then we can write the series as

Way 2: Pull out from each term and rewrite the series as

This can be viewed as a series with general term

starting at *n* = 2. We write the series in summation notation as

This is equivalent to what we got with Way 1.

(c) Way 1: In order to start at *n* = 1, we can break up the denominators like this:

Then we can write the series as

Way 2: This is very similar to (b), except that we'll pull out from each term so we can view the remaining powers of 2 as starting at *n* = 1.

(d) Way 1: Rewrite the series so the denominator of term *a*_{n} has a factor of 2^{n}, starting at *n* = 0:

Then we can write the series as

Way 2: If we want the first term to involve 2 with an exponent of 0, we need to factor out from every term: