Is it possible to change this into a series?

[Damoo]

(x^3+x+3) + 1/x + -1/x+2


where x=1 all the way to 99?

Suppose to turn into a telescoping series..

Any help appreciated.

Thanks,


Original question was

x^5+2x^4+x^3+5x^2+6x+2/x^2+2x

I had to do an polynomial division and decomposed into a partial fraction to get to this point.
The division works out to x^3+x+3 with a remainder of 2.

My work on the partial fraction:

(x^3+x+3) + 2/x^2+2x

2/(x^2+2) = A/x + b/x+2

2= A(x+2) + B(x)

Let x= -2
2=a(0) + b(-2)
B= -1

Let x=0
2=a(2) + b(0)
A= 1

2/x(x+2) = 1/x + -1/x+2

 

[lookagain]

(x^3+x+3) + 1/x + -1/x+2


where x=1 all the way to 99?

Suppose to turn into a telescoping series..

Any help appreciated.

Thanks,


Original question was

x^5+2x^4+x^3+5x^2+6x+2/x^2+2x

I had to do an polynomial division and decomposed into a partial fraction to get to this point.
The division works out to x^3+x+3 with a remainder of 2.

My work on the partial fraction:

(x^3+x+3) + 2/x^2+2x

2/(x^2+2) = A/x + b/x+2

2= A(x+2) + B(x)

Let x= -2
2=a(0) + b(-2)
B= -1

Let x=0
2=a(2) + b(0)
A= 1

2/x(x+2) = 1/x + -1/x+2

1) Use grouping symbols.

2) Spread out your text/characters, as they are difficult to read/distinguish.

3) Use the same letter consistently for the same variable. Here,
type "B," not "b" and "A," not "a," wherever you are using that for
the coefficients in the partial fraction method.


The way I would have presented it:

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Original question was

(x^5 + 2x^4 + x^3 + 5x^2 + 6x + 2)/(x^2 + 2x)

I had to do an polynomial division and decomposed into a partial fraction
to get to this point. The division works out to x^3 + x + 3 with a remainder of 2.


My work on the partial fraction:

(x^3 + x + 3) + 2/(x^2 + 2x)

2/(x^2 + 2x) = A/x + B/(x + 2)

2 = A(x + 2) + B(x)


Let x = -2:

2 = A(0) + B(-2)

B = -1


Let x = 0:

2 = A(2) + B(0)

A = 1


2/[x(x + 2)] = 1/x - 1/(x + 2)[/QUOTE]

 

[lookagain]

(x^3+x+3) + > > > 1/x - 1/(x + 2) < < <


where x=1 all the way to 99?

Suppose to turn into a telescoping series..

The portion of your expression, 1/x - 1/(x + 2) taken from x = 1 to 99, is a telescoping series:


1/1 - 1/3 + 1/2 - 1/4 + 1/3 - 1/5 + 1/4 - 1/6 + 1/5 - 1/7 + 1/6 - 1/8 + 1/7 - 1/9 + ...

+ 1/96 - 1/98 + 1/97 - 1/99 + 1/98 - 1/100 + 1/99 - 1/101 =



1 + 1/2 + ... - 1/100 - 1/101 = ?