a) Determine if the series 1 1=1 n(n+1) is convergent. If it is convergent, find the sum b) Determine the Maclaurin series for In(1 + x) c) What is
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hey could someone help me with these and show full working so i can follow along? im having alot of trouble with

them  a) Determine if the series
1
1=1
n(n+1)
is convergent. If it is convergent, find the sum
b) Determine the Maclaurin series for In(1 + x)
c) What is the radius and interval of convergence in (b)?
d) Show that the Maclaurin series for (1 + x) In(1 + x) is
(-1 )ntlyn+1
n(n + 1)
e) Hence find lim (1 + x) In(1 + x) . Use your answer to determine lim x In(x). What is
X--1+
lim x* ?
x-0+
f) Use the first two non-zero terms of the Maclaurin series in (d) to approximate (1.1) In(1.1).

View the full answer a) By Raabe's test for series , we know that ,
if IUn be a series of the great number
and limn ( un - 1 ) - 1 .
Them 5 Un convergent it lyl . &amp;
Iun divergent if !(1.
So ; let , un-
n (nt ! )... the sum of the series in = 1.
b )
Maclurian series : -
+ ( x ) =
p ()
&quot;) (0 )
o !
m=0
Now
. f ( &quot;) = In (1+2 ) .. f ( 0 ) = 0
I' ( &quot; ) = +
. : I' (0 ) = 1
8 ( 21 ) = - 1
(1 71 ) 2
.:... - 1
- .' lim
It In
1= 1
. .. R = radius of convergent of
in
J .
This series converge far
121 &lt; 1
= 1
-1&lt; &lt; 1
If; a=1 . .
This' series is alternating series
and
In is monotonically... n
in
divergent
&amp;
The series M Aimnexgen
GIm+ 1
n
ios
in divergent
fan
2 =-1
The interval of convergence in
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