Finding Intervals of Convergence In Exercises 49-52, find the intervals of convergence of (a). f ( x ) (b) f ' ( x ) , (c) f " ( x ) . and (d) ∫ f ( x ) d x . . (Be sure to include a check for convergence at the endpoints of the intervals.) f ( x ) ∑ n = 0 ∞ ( x 3 ) n
Finding Intervals of Convergence In Exercises 49-52, find the intervals of convergence of (a). f ( x ) (b) f ' ( x ) , (c) f " ( x ) . and (d) ∫ f ( x ) d x . . (Be sure to include a check for convergence at the endpoints of the intervals.) f ( x ) ∑ n = 0 ∞ ( x 3 ) n
Solution Summary: The author calculates the Interval of Convergence of la)f(x), b)
Finding Intervals of Convergence In Exercises 49-52, find the intervals of convergence of (a).
f
(
x
)
(b)
f
'
(
x
)
, (c)
f
"
(
x
)
. and (d)
∫
f
(
x
)
d
x
.
. (Be sure to include a check for convergence at the endpoints of the intervals.)
4.
Find a power series, centered at c = 1, for the following function:
2
f (x) =
Definition We say that a sequence {fn(x)} of functions converges pointwise to a function fo(x) on
an interval I if
lim fn(xo) = fo(xo)
n-00
for each xo E I.
Let fn(x) = x" for all r e [0, 1]. Then {fn(x)} converges pointwise to a function fo(x) on the
interval [0, 1] where
A for x E [0, 1)
B for x = 1
fo(x) =
a)
What is A? What is B?
b)
Which of the following statements is true.
i) If {gn (x)} is a sequence of continuous functions that converges pointwise on [0, 1] to go(x),
then go(x) must also be continuous on
[0, 1].
ii) If {9n(x)} is a sequence of differential functions that converges pointwise on [0, 1] to go(x),
then go(x) must also be differentiable on [0, 1]
iii) Both i) and ii) are true.
iv) None of the above.
(x + 2)"
n2(-3)"
Find the interval of convergence
for
|
n=1
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