How do I go about proving limits for given sequences and series, for example how would i prove that the limit when x tends to infinity of n^(1/n)=1?

How do I go about proving limits for given sequences and series, for example how would i prove that the limit when x tends to infinity of n^(1/n)=1?

Let f(x)= ∑∞n=0 (x^n)/(n!) Find f(0)= 1 f(1)= e f'(x)= f(x) f''(x)= f(x) Not exactly sure how we're supposed to go about finding those answers, or how to start.

Given numbers 01 to 49, make unique series of 6 numbers....I e if we select 5,11,19,22,31,49/1,9,16,29,33,44 then they shd be found there.....

Please note that I am talking about a series and NOT a sequence

I couldn't figure out how to do this :/

an=e(-1/sqrt(n)) Apparently in converges and the limit is 1 but I'm not really sure how to do this type of problem with a limit of ex {n2e-n} Limit converges to 0? How?...

Let (an) and (bn) be sequences of real numbers with an ≥ 0 for all n ∈ N. Suppose that an ≤ bn for all n ∈ N and that the sum of all bk from k=1 to infinity is convergent. Prove...

A series is given as below, 2-2^2/(x+1)+2^3/(x+1)^2 -2^4/(x+1)^3 +⋯+2^r/〖(-(x+1)〗^(r-1) +⋯ I) Find the set of values of x such that the series converges...

FORMULAE FOR SERIES +N/2+N/4+N/8......1

Can anyone explain how you prove a general term is true for a series, un+1=un/1+un u1=2 (i) find u2 and u3 (ii) Hence suggest an expression for un and so I have found...

pls clear my confusion

∞∑n=1 (x−4)^n /n^n

1,3,7,?,121,721

abbcccddddeeeee........... what will be the 126th digit?

Determine whether the sequence is increasing, decreasing, or not monotonic. an = 1/(6n+7) Is the sequence bounded? Yes or No? Explain.

Determine whether the sequence is increasing, decreasing, or not monotonic. an = 4n(−7)n Is the sequence bounded? Yes or No? Explain.

A. summation of negative forty five times n from n equals zero to fifteen B. summation of the quantity negative nine plus five n from n equals zero to fifteen C. summation of the quantity...

A ball is thrown vertically upwards from the ground. It rises to a height of 10m and then falls and bounces. After each bounce it rises vertically to 2/3 the height from which it fell. (i)...

The first 3 terms of a geometric expression are x, 10 − x ,and 2x + 1 where x > 0. Find the value of the common ratio, r.

I have a series (Un)n∈Ν. U0=2 and Un+1=1/4(Un)+1. I need to find a series (Vn)n∈Ν and a∈R such that Vn=Un+a and that (Vn) is geometric.