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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them2.png

2.png

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).
Check your answer with a calculator.

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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 . &
Iun divergent if !(1.
So ; let , un-
n (nt ! )...

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the sum of the series in = 1.
b )
Maclurian series : -
+ ( x ) =
p ()
") (0 )
o !
m=0
Now
. f ( ") = In (1+2 ) .. f ( 0 ) = 0
I' ( " ) = +
. : I' (0 ) = 1
8 ( 21 ) = - 1
(1 71 ) 2
.:...

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- 1
- .' lim
It In
1= 1
. .. R = radius of convergent of
in
J .
This series converge far
121 < 1
= 1
-1< < 1
If; a=1 . .
This' series is alternating series
and
In is monotonically...

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n
in
divergent
&
The series M Aimnexgen
GIm+ 1
n
ios
in divergent
fan
2 =-1
The interval of convergence in
(-1, 17

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