LAB 31 11.10 TAYLOR & MACLAURIN SERIES Lab Time: ____________ Date: __________
Basic Definitions:
a
Using the known Geometric Series Formula: ar
n0
n
a ar ar 2 ar 3 ...
1 r
which holds when r 1,
new series formulas can be created by a suitable replacement of terms.
a
a ar ar ar ... ar and making the replacements a 1
2 3 n
1. Using the formula
1 r n0
and r x , create a new series formula for 1 and give its interval of converegence.
1 x
1
1 1x 1x 2 1x 3 ... 1x n
1 x n 0
1
1 x x x ... x
2 3 n
1 x n 0
r 1 x 1 1 x 1
Interval : 1,1
1
2. Replace the x-term in your formula from the previous problem with x to create a formula for .
1 x
1 1
1 x x 2 x 3 ... x n
1 x 1 x n 0
1 x x 2 x 3 ... 1 x n
n
n0
3. Integrate your formula from the previous problem with term-by-term to create a formula for ln x 1 .
(Don’t forget to find the constant “C” by plugging in x 0 )
ln x 1 x 11 dx 1 x x2 x3 ... dx C x 1
2 x 2 13 x3 1
4 x 4 ...
ln x 1 C x 1
2
x 2 13 x3 1
4
x 4 ...
x 0 ln 0 1 C 0 0 13 0
3 2
1
2
1
4 04 ...
ln 1 C 0 0 0 0 ...
0 C
ln x 1 0 x 1
2
2 3
x 13 x 1
4
4
x ...
x4 ... 1n
n1
So : ln x 1 x 1
2
2
x x 1
3
3 1
4 xn
n 1
1
4. Replace the x-term in the formula 1 x x 2 x3 ... x n
1 x n 0
1
with x2 to create a formula for .
1 x 2
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