Real Analysis Answers

Test the series: infinity sigma n=1 (-1)^ n-1 sin nx n sqrt n for absolute and conditional convergence

Give an example of a series sigmaan such that Sigma an is not cnvergent but the sequence (an) converges to 0,

For the function f(x)= x^2-2 defined over[1,6],

Verify L(P,f)≤U(P,f) where is the partition which divide [1,6] into five equal intervals

prove that the function f given by

f(x)={ 2,if x is irrational

-2,if x is rational

is discontinuous, for all x€R,using the sequential definition of continuity

Is there a continuous function f:[0,1]~>[0,1] that is not constant in any nontrivial interval such that f^-1{0} is uncountable?

Let {an} sequence is defined as a1=3, an+1=1/5(an)

converges to zero. Prove that

show that

a) lim x- infinity (x-3/x-1)^x =1/e^2

b) lim x->5/3 1/(3x+5)^2 =infinity

Examine the function ,f(x)=(x+1)^3(x-3)^2 for extreme values

prove that lim n->infinity [1/√2n-1 +1/√4n-2^2 +1/√6n-3^2 +.......+1/n]=π/2