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.