Determine Whether series converge or diverge if it converge what is the limit.  $\{ \frac {(-1)^{n-2}n^{2}}{4+n^{3}}\} _{n=0}^{\infty }$

Use geometric series formula to calculate: 10+50+250+...+10*5^m, where m(greater than or equal to)0

Find the geometric series or state that it diverges for n=0,infinity        e^-6n

1+x+x^2+x^3+...+x^k determine the sum of the geometric series

I have done a bunch of these of problems, but this one step looks wrong in the solution for this problem.  https://www.bartleby.com/solution-answer/chapter-62-problem-71e-calculus-volume-2-17th-edition/9781938168062/in-the-following-exercises-express-each-series-as-a-rational-function-71-n11-x3-2n1/b209f360-2097-11e9-8385-02ee952b546e On the very first step when I distribute the (x-3)^(2n-1) exponent in the denominator to be (x-3)^2n and (x-3)^-1, the (x-3)^-1, should flip up to the numerator (not stay in the denominator as (x-3)), but it doesn't according to the answer key.  And the answer is right but that first step LOOKS wrong.  Can you help explain that exponent distribution?

a_{n}=\frac{(-3)^{n}}{n !}

Turn summation to infinity with n=1 of ((-1)^n•x^(n))/n into a power series

Determine whether the alternating series E (-1)^+1° converges or diverges. n = 1 n3

Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.) an = n^3/ 3n + 9lim n→∞ an =

Please explain why sequence Xnk approaches +infinity and why sequence Xmk approaches -infinity. Why would one use each of the sequences if the main sequence is (n sin n)? Lastly, can I assume sequence sequence Nk sin Nk <1/2 ...therefore making it convergent? Why the other way?