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| 2n-1) 2. Consider the sequence |(n+1)! a) is the sequence monotone increasing or monotone decreasing or neither? b) Find upper and lower bounds for the sequence. c) Does the sequence converge or diverge? (Explain) 3. Determine if the series converges or diverges. If it converges, find its sum. => [-1-] c) Ë ?j? – 1-1 j? +1

Answer 1

2)

n=1 2n-1 (n+1)!

1-27 ا 2 = 7 .ج امانت (n+1)

n=3= 2n-1 (n + 1)! 4 =

Which is continuously decreasing

In general we can say that for all natural numbers n

21- 1 (n+1)! 20 (n + 2)! as n +2 > 2

So the sequence is monotonically decreasing

b)
f(1) = 0.5
is upper bound and is lower bound (as values are always positive)

c) It converges because is bounded below and continuously decreasing. It can be proved that the sequence converges to 0

$\blacksquare$

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