234k views
1 vote
If the series converges, find the sum of the series: ∑(7 * 2ⁿ) from n=1 to [infinity]. (If the series diverges, leave this box blank.)

a) 14
b) 21
c) 28
d) 35

User Der
by
8.0k points

1 Answer

4 votes

Final answer:

To find the sum of the series ∑(7 * 2ⁿ) from n=1 to ∞, we can use the formula for the sum of a geometric series. The sum of the series is -7.

Step-by-step explanation:

To determine if the series ∑(7 * 2ⁿ) from n=1 to ∞ converges, we need to first recognize the type of series we are dealing with. This series is a geometric series, where each term is found by multiplying the previous term by a constant.

The general form of a geometric series is: ∑(a * rⁿ) from n=0 to ∞ where: - 'a' is the first term of the series, - 'r' is the common ratio between the successive terms. For our series, the first term 'a' when n=1 is 7 * 2^(1-1) = 7 * 1 = 7. The common ratio 'r' is 2, since each term is twice the previous term.

A geometric series converges if and only if |r| < 1. Since our common ratio r = 2, which is greater than 1, the series does not converge.

For geometric series where |r| < 1, the sum to infinity is given by the formula: S = a / (1 - r) However, since our r is not less than 1, this formula does not apply, and the series diverges. Therefore, we do not have a sum for the series, and the box should be left blank.

User Min Min
by
8.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories