Applied Dutch math book 2

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Question:Applied Dutch math book 2

Posted: The function 110 Product: Maple

December 19 2021

Hello everybody,

I managed to get to the point that i could start part 2 of the book series of applied Dutch math.

This paragraph is about Taylor series.

Now i am being asked to find a solution for the taylorseries with a certain x value. That is all fine, Maple can spit it out. But to put in into something that is there with a sum sign in front of it is something else. I have to convert the solution into something that is written with "x to the k-ed, something something".

Now i did find to solution to a not so complex one. I will add that one on the maple sheet. But there is this one that is really complex, and i cant get my head around how to get to the final solution that makes the sum go and work well. It does have some series to it. But i dont know how to find that one out. Is there some trick to make it work?

Without further ado:

(1)

Sum((x-2)^k, k = 0 .. infinity)

(2)

series(1+(1/2)*(x-1)-(1/8)*(x-1)^2+(1/16)*(x-1)^3-(5/128)*(x-1)^4+(7/256)*(x-1)^5-(21/1024)*(x-1)^6+(33/2048)*(x-1)^7-(429/32768)*(x-1)^8+(715/65536)*(x-1)^9-(2431/262144)*(x-1)^10+(4199/524288)*(x-1)^11-(29393/4194304)*(x-1)^12+(52003/8388608)*(x-1)^13-(185725/33554432)*(x-1)^14+(334305/67108864)*(x-1)^15+O((x-1)^16),x = 1,16)