Use Exercise, to find the limit of each sequence.(a) (b) (c) (d) (e) (f) Let (sn) be the s...

Use Exercise, to find the limit of each sequence.

(a)

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(b)

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(c)

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(d)

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(e)

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(f)

Let (sn) be the sequence defined by sn = (1 + 1/n)n. Use the binomial theorem (Exercise) to show that (sn) is an increasing sequence with sn< 3 for all n. Conclude that (sn) is convergent. The limit of (sn) is referred to as e and is used as the base for natural logarithms. The approximate value of e is 2.71828.

Define the binomial coefficient  by

 for r = 0, 1, 2, n.

(a) Show that  for r = 1, 2, 3, n.

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(b) Use part (a) and mathematical induction to prove the binomial theorem: