Final answer:
The limit of the given geometric series is 77.
Step-by-step explanation:
The given series, ∑ₙ=1[infinity](7²-4²)(4/7)ⁿ, is indeed a geometric series. A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is (4/7). To find the limit of the series, we can use the formula for the sum of an infinite geometric series:
S = a/(1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio. The first term of the series is (7²-4²) = 33, so substituting the values into the formula:
S = 33/(1 - 4/7)
Simplifying the expression:
S = 33/(3/7)
S = (33 * 7)/3
S = 231/3
S = 77
Therefore, the limit of the given geometric series is 77.