If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Question

∑n=1 to ∞

1+7^n/ 8^n



Answer 1

We see that the terms are all positive. If nth term = a_n = 1 + (7/8)ⁿ , then:

lim_n→∞ a_n = lim_n→∞ [1 + (7/8)ⁿ] = 1 + 0 = 1 ≠ 0

SInce the terms of the series are all positive and do not approach 0 as n approaches infinity, the series diverges by the nth term test.

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If nth term = (1 + 7ⁿ) / 8ⁿ then we have:

∑_{n=1 to ∞} [(1/8)ⁿ + (7/8)ⁿ] = ∑_{n=1 to ∞} (1/8)ⁿ + ∑_{n=1 to ∞} (7/8)ⁿ

Both series are convergent geometric series, the first with r = 1/8 and the second with r = 7/8.

So, the series converges and has sum = (1/8) / [1 - 1/8] + (7/8) / [1 - 7/8] = 1/7 + 7 = 50/7