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matrix_204

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**Convergence tests?**

I was having some trouble deciding which convergence tests to use for some of the following problems, as i have about a day or less to work on them. So please just tell me which convergence tests are easiest in doing these problems and some tips as sum of them require more than normal time to do.

1)a) let [S_k]_k>=1 be a sequnce with positive integers in increasing order and that do not have a 6 in their decimal expansion. For example, the sequence begins as; 1,2,3,4,5,6,7,8,...,15,17,...59,70,...

Prove carefully that the (sum of the series 1/N_k as k goes from 1 to infinity) converges to a number less than 82.

b) Now for the other part, make a sequence(say {M_k}_k>=1) such that it contains positive integers in increasing order and whose decimal expansions end in 6.

Now prove that this series diverges. (i.e. sum of 1/M_k as k goes from 1 to infinity)

2. Let a, b >=0, prove that the lim of (a^n + b^n)^1/n +max{a,b} as n goes to infinity.(this one doesn't seem very difficult and i have yet to do this one, but not much to worry about)

3.i) This problem i kinda had an idea but i think i lost it, i had this problem given long ago, but it goes like this, prove that if 0<a<2 then a<root2a<2. (this part of the problem doesn't seem to relate to convergence but i guess it leads to the second part).

ii) Prove the sequence {root2, root(2root2),root(2root(2root2)),...} converges. And what is its limit?

(so what i think u have to do here is, define a sequence, say {a_n}_n>=0 recursively (btw would someone help me understand how you define recursive formulas, i mean I've done some problems where u are given certain sequences but i don't kno how to come up with a formula, so if anyone knows an easy way to make me understand this, please mention it).

let a_0=root2 and a_n+1=root(2(a_n)), then a_n converges.)