Determine whether the series converges or diverges. 1) the semation to infinity

Question

Determine whether the series converges or diverges. 1) the semation to infinity of 1/k^7 , k=1 2)the semation to infinity of (k+1)/k^3 , k=1 3) the sum of the semation to infinity of k/(k-2)^5 , k=1 4)the semation to infinity of 1/(k+3) , k=1 5)the semation to infinity of 5cosk(pie) , k=1 6)the semation to infinity of 6^k /5 , k=1

Explanation / Answer

1), 2), and 3) all converge to zero. The power of k in the denominator is greater than 1, that is, (1/k^p) has p>1. 4), 1/k has p=1. This converges to zero (the series converges), but the summation does not exist. 5) cosine oscillates as k increases, so this diverges. 6) the (1/5) can be pulled outside the sum so consider the summation for 6^k. 6^k grows exponentially, so the terms in this series diverges to infinity.

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