Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that has a power series expansion. Do not show that R(x) → 0.] Find the associated radius of convergence R.
f(x) = 2(1-x)-2
Step 1
The Maclaurin series formula is f(0) + F'(0)x+
f(x) = 2(1-x)-2 has the following derivatives.
f'(x) = 4✔
f"(x) = 12✔✔
F"(x) = 48✔
Step 2
Now, f(0) = 2✔
Step 4
f(4)(x) = 240✔✔
Step 6
Step 3
Therefore, the Maclaurin Series begins as follows.
+6✔
+4✔
tot
+10✔
Thus, the general formula is
lim
an +1
an
= lim
F'(0) = 4✔
Simplifying, we get |x|lim
F"(0) 2
2!
-4✔
Step 5
Using the Ratio Test to find the radius of convergence, we have
2(n + 2)x+1
0+ 0 +1
2(n+1)x²
+2 = |x| · |
2(n+1)
+
n+1
F(0)3 + F(4)(0) 4
F"(0) = 12✔
2(n+1)x"
2(n + 1)
X
13
8x²
f(0) = 48
48 and f(4)(0) = 240
240](https://content.bartleby.com/qna-images/question/86624ab7-da2a-4a5b-91b8-0bf4ad8ab366/821cb23d-00a5-441a-8ca1-620d60e1baef/iqwg7f_processed.png)
Transcribed Image Text:Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that has a power series expansion. Do not show that R(x) → 0.] Find the associated radius of convergence R.
f(x) = 2(1-x)-2
Step 1
The Maclaurin series formula is f(0) + F'(0)x+
f(x) = 2(1-x)-2 has the following derivatives.
f'(x) = 4✔
f"(x) = 12✔✔
F"(x) = 48✔
Step 2
Now, f(0) = 2✔
Step 4
f(4)(x) = 240✔✔
Step 6
Step 3
Therefore, the Maclaurin Series begins as follows.
+6✔
+4✔
tot
+10✔
Thus, the general formula is
lim
an +1
an
= lim
F'(0) = 4✔
Simplifying, we get |x|lim
F"(0) 2
2!
-4✔
Step 5
Using the Ratio Test to find the radius of convergence, we have
2(n + 2)x+1
0+ 0 +1
2(n+1)x²
+2 = |x| · |
2(n+1)
+
n+1
F(0)3 + F(4)(0) 4
F"(0) = 12✔
2(n+1)x"
2(n + 1)
X
13
8x²
f(0) = 48
48 and f(4)(0) = 240
240
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