does the series converge?. if so find limit......Is the sequence incresing or decrease or monotonic..?

Question

{ (-8)^n/ n!}

Answer 1

This series is absolutely convergent, because lim |a(n+1)/a(n)| = lim (8^(n+1))/8^n * n/(n+1)! = lim 8/n = 0 < 1. So, it is convergent.

The sequence { (-8)^n/ n!} is neither increasing nor decreasing, it oscillates.


But finding the limit of the series doesn't seem easy, I'll think about it.

Answer 2

The series for e^x is sum (n=0 to ∞, x^n / n!) and converges for all x, including x = -8 as in your example.

If an is the nth term, regardless how big the absolute value of x is, there will eventually be a value n > x. Until that point, the sequence will diverge, but after n > x, the series will start converging faster and faster as the ratio x/n becomes smaller and smaller. Because the sign of each term alternates with the one before it, this particular series will oscillate about its final value of e^(-8).