Calc 2 question please, please help me!?

Let
S(x) =
sum from n=0 to ∞ of cnx^n
be a power series with radius of convergence R, and suppose that the power series
S(x)
converges when
x = 3
and diverges when
x = −7.
Which of the following conclusions must necessarily be true? (Select all that apply.)

The series c0 − c1 + c2 − c3 + converges.

R ≥ 3

R < 7

The series c0 − 3c1 + 9c2 − 27c3 + converges.

The series c0 + 7c1 + 49c2 + 343c3 + diverges.

The series −c0 + 9c1 − 81c2 + 729c3 + diverges.

The series c0 + 4c1 + 16c2 + 16c3 + diverges.

None of the conclusions must necessarily be true.

1 Answer
Ananda Dasgupta
Apr 17, 2018

See below.

Explanation:

If the radius of convergence of a power series

#c_n(z-z_0)^n#

is #R#, then

  • the series definitely converges for #|z-z_0| < R#
  • the series definitely diverges for #|z-z_0| > R#
  • may or may not converge for #|z-z_0|=R#

We consider

#f(x) = sum_{n=0}^oo c_nx^n#

Here, we have #x_0=0# .

Since the series converges for #x = 3#, we have #|x-x_0| = 3 <= R#

#color(red)(R >=3)#

Since the series diverges for #x = -7#, we have #|(-7)-0|=7>= R#. Thus, the stated conclusion is not correct

#color(blue)(R<= 7)#

(The conclusion, as written, asserts that #R < 7# )

The series #-c_0+9c_1+9c_2-27c_3+...# is #f(-3)#. Since #|(-3)-0| = 3# and this could be the value of #R#, we can not be sure that the series necessarily converges.

The series #c_0-7c_1-81c_2+729c_3+...# is #f(+7)#. Since #|7-0| = 7# and this could be the value of #R#, we can not be sure that the series necessarily diverges.

The series #-c_0+9c_1-81c_2-729c_3+...# is #-f(-9)#. Since #|(-9)-0| = 9# and this is certainly #> R#, the series necessarily diverges.

The series #c_0+4c_1+16c_2+64c_3+...# is #f(4)#. Since #|(4)-0| = 4# and this could be the value of #R#, we can not be sure that the series necessarily converges or diverges.

(It seems that there is a typo in the question - the coefficient of the last series is stated as 16 and not 64).

If on the other hand we are talking about the series #c_0+2c_1+4c_2+8c_3+...#, we would have been sure that it converges, since #|2-0|=2 < R #.

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