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.)
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.
9.
The graph of r = 2 sin 30 is shown to the right.
Write an integral which can be used to find the
area of one of the "petals". Calculate and
determine what that area is.
(x + 2)"
n2(-3)"
Find the interval of convergence
for
|
n=1
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