Weekly Assignment #7

MAT 229, Spring 2021

Instructions:  Show your work!

1. Recursively defined sequence

Determine whether the sequence defined as follows is convergent or divergent.
	(Subscript[a, 1])=1
(Subscript[a, n])=4-(Subscript[a, n-1]) for n>1
If it converges, what does it converge to. If its diverges, describe how it diverges. For example, does it converge to ∞ or -∞? Does it stay bounded? Does it oscillate?

Answer the same questions for this same recursive definition (Subscript[a, n])=4-(Subscript[a, n-1]) but with first term (Subscript[a, 1])=2

2. Geometric series

Find the value of b such that (((Underoverscript[Σ, (k=1), ∞])((1/(1+b))))^k)=2.

Express the repeating decimal number 0.467467467…. as a ratio of integers by first writing it as a geometric sum.

Repeat the second part with the repeating decimal number 0.999.... What’s the surprise?

3. Partial sums

If the (n^th) partial sum of series (Underoverscript[Σ, (k=1), ∞])(Subscript[a, k])=(Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+… is (Subscript[S, n])=3-(n/(2^n)). Does the series converge? If so, to what value?

Write the first four partial sums of (Underoverscript[Σ, (k=1), ∞])((k/(2^k))-((k+1)/(2^(k+1)))). What is the form of (Subscript[S, n])? Does the series converge? If so, to what value?