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1. There exist real numbers A and B so that $$\frac{1}{k(k + 3)} = \frac{A}{k} + \frac{B}{k + 3}.$$
for all real numbers k other than 0 and -3 Enter the ordered pair (A, B).

2. What is $$\sum_{k=1}^{\infty} \frac 1{k(k+3)}?$$

3.  Compute the sum and enter your answer as a common fraction $$\begin{array}{r r@{}c@{}l} & 1 &.& 11111111\ldots \\ & 0 &.& 11111111\ldots \\ & 0 &.& 01111111\ldots \\ & 0 &.& 00111111\ldots \\ & 0 &.& 00011111\ldots \\ & 0 &.& 00001111\ldots \\ & 0 &.& 00000111\ldots \\ + &&\vdots \\ \hline &&& ~~~? \end{array}$$

4. Find f(k) such that $$\sum_{k=1}^n f(k) = n^3.$$

Dec 29, 2019

#1
-2

1. By partial fractions, (A,B) = (2/3,-4/3).

2. The sum telescopes to 3/4.

Dec 29, 2019
#2
+119
0

Dec 29, 2019
#6
+108734
+1

Why do you think your rebuke is worth 3 points NewMember?

I think it is worth -3 points.

Behave yourself and say thanks for the person's interest before you suggest that an anwer is wrong.

AND

If you believe an answer is wrong then you are expected to say why you know this to be true.

e.g. the answer in the book is .....

Melody  Jan 6, 2020
#3
+109563
+2

1)

1                     A               B

_______  =      _____  +   _____                       multiply through by  k (k + 3)

k(k+3)               k             k +  3

1  =   A(k + 3)   + Bk        simplify

1 +  0k  =   ( A + B)k  +  3A

Equating like terms

3A  =   1

A = (1/3)

And

A + B  =  0

B = -1/3

So

(A, B)    =  (1/3, - 1/3)

Dec 29, 2019
#4
+109563
+2

3)

We  actually have the  series

(1 + 1/9)    +  (1/9)  +   (1/90)   + 1/900  + ......+     =

(10/9)   +   (1/9)    +  (1/90) +  1/900

The common ratio, r,  =    (1/10)    and the first term  is  (10/9)

So....the sum of this infinite series   =

(10/9)              (10/9)               (10/9)            10       10           100

______   =     ________   =    ______  =      ___ *  ___   =   _____

1   -  r              1  -1/10            (9/10)             9         9             81

Dec 29, 2019
#5
+108734
0

Attn:   NewMember.

You ask four questions all at once which automatically means you are using this site as a cheat site.

You give yourself as many points as is humanly possible. (it is in really bad taste to give yourself points for posting questions)

You criticize other people when you do not like their answers.

You take points off people at the drop of a hat.

And you never say thank you to anyone!