Answer
(a) We can rewrite the series so the terms are fractions whose denominators are powers of 2:
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_20.png)
From here it makes sense to view the series as starting at n = 3 and having general term
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_21.png)
Then we can write the series in sigma notation as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_22.png)
(b) Way 1: We already saw that we can rewrite the series as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_23.png)
In order to think of the first term as using n = 2 instead of n = 3, we can break up the denominators like this:
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_24.png)
Then we can write the series as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_25.png)
Way 2: Pull out
from each term and rewrite the series as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_26.png)
This can be viewed as a series with general term
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_27.png)
starting at n = 2. We write the series in summation notation as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_28.png)
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:
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_29.png)
Then we can write the series as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_30.png)
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.
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_32.png)
(d) Way 1: Rewrite the series so the denominator of term an has a factor of 2n, starting at n = 0:
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_33.png)
Then we can write the series as
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_34.png)
Way 2: If we want the first term to involve 2 with an exponent of 0, we need to factor out
from every term:
![](https://media1.shmoop.com/images/calculus/calc_series_defn_latek_35.png)