$$f ( x ) = sum _ { n = 0 } ^ { + infty } frac { ( – 1 ) ^ { n } ( 2 n + 3 ) } { n ! } x ^ { 2 n }$$

Please help to tell me the methods to find a closed form of the infinite series.

Thank beforehand!!

# Tag: series

## How to calculate Laurent series in mathematica about some point zo and annulus a

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## What is the probability of winning a best of 5 series with varying probability in each game?

Suppose two friends, A and B, play a best of 5 (first to 3 wins) of 5 different games. Due to skill levels of each player depending on the game, each game has a different probability for each player to win. The outcome of each game is independent of the outcomes of the others.

Label games as G1, G2, G3, G4, G5 (not necessarily played in order).

P(A wins G(i))=p(i)

What is the probability that player A wins the series?

(for the sake of my question, p(1)=0.35, p(2)=0.7, p(3)=0.55, p(4)=0.5, and p(5)=0.4)

Thanks in advance!

## complex analysis – How to write this series as a ratio of polynomials?

I have the series:

$g(z) = sum^{infty}_{0} b_{n}z^{n}$ where $zin$$mathbb{C}$.

$b_{n}$ is the $n^{th}$ number in the Fibonacci sequence, i.e. $(b_{0},b_{1},b_{2},b_{3},…)$ = (1,1,2,3,…)

How can I write this expression as a ratio of polynomials?

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Interview: Q&A with RackNerd CEO Dustin B. Cisneros on Leading, Learning, Help and Execution

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## sequences and series – How to find the minimum number to be multiplied to k so that it’s last digit is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9?

How to find the minimum number to be multiplied to k so that the last digit of the product is k?

for example if K = 7, then then least number to be multiplied to k to get last digit is 0, 1, 2, 3, 4, 5, 6, 7,

8, 9 are respectively { 10, 3, 6, 9, 2, 5, 8, 1, 4, 7 };

## Series descending to less than 1

Given a large number, say 6.028033E17, how many times must this be cut in half to become less than 1?

## sequences and series – Can anyone explain this, please?

I “discovered” the following when I began fooling around in my head with the numbers on my digital clock. Pretty simple, really: 12:45, for example can be added this way – 1+2+4+5= 12. Now, add the 1+2 and get 3. (Remember the 3 because you’ll see it again.) Now add 12:45 another way. 12+45=57 and 5+7=12 and 1+2=3 – again. Now try 24+15=39 and we’re back to the 12 and 3 again. You can do 124+5=129=12=3 again. No matter how you mix these four numbers up, add them together and then add the sums down to a single digit, you always wind up with 3. This works for huge numbers as well as small ones. If you create a random assortment of numbers – for example, 4739251683902165 and add them together sequentially you get 71. 7+1=8. If you divide them into more than one grouping to add together, each of those groupings will produce a different final number but, when added togrther, those final numbers will equal 8. For example 4739+2165=6904 and 6+9+0+4=19 and 1+9=10 and 1+0=1. Now add the other two groups together – 2,516+8,390=10,906. 1+0+9+0+6=16 and 1+6=7. Add the 1 from the first grouping to the 7 from the second grouping and you’re back to another 8. I’m certain there is a name for this seemingly odd occurrence and a reason it should be. Does anyone know where I might find some information about it? Thank you.

## Can i find a power series expansion of this function without a taylor series?

I was asked to find the power series expansion of f(x) = x/√(4+x^2) about x = 0. Is there a way to do a power expansion without finding the Taylor series? Deriving this function multiple times seems extremely tedious.

## analytic number theory – The Dirichlet series of trigonometric functions

I am searching a question now .In that question i think some ideas. i am really interested in if we can find the dirichlet series of trigonometric functions like sin,cos,tan ,cot or not? if there are ; what kind of trigonometric functions are ?

i think maybe its a research question.So i think my question is appropriate for MO. But if its not, i can delete my question and ask for MSE .Thanks for your answers.