Use sigma notation to write the series
starting at (a) n = 3, (b) n = 2, (c) n = 1 and (c) n = 0.
(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:
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 an has a factor of 2n, starting at n = 0:
Way 2: If we want the first term to involve 2 with an exponent of 0, we need to factor out from every term: