Infinite Series of Functions - Introduction to Analysis
Card 0 of 4
Determine whether the following statement is true or false:
Some power series such as
have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
Determine whether the following statement is true or false:
Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When
is fixed,
, and 
with the assumption that
and
the Root Test is applied to the series
.
When
by the assumption and the notation that

implies that
therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When is fixed,
, and
with the assumption that and
the Root Test is applied to the series
.
When by the assumption and the notation that
implies that therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
Compare your answer with the correct one above
Determine whether the following statement is true or false:
Some power series such as
have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
Determine whether the following statement is true or false:
Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When
is fixed,
, and 
with the assumption that
and
the Root Test is applied to the series
.
When
by the assumption and the notation that

implies that
therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When is fixed,
, and
with the assumption that and
the Root Test is applied to the series
.
When by the assumption and the notation that
implies that therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
Compare your answer with the correct one above
Determine whether the following statement is true or false:
Some power series such as
have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
Determine whether the following statement is true or false:
Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When
is fixed,
, and 
with the assumption that
and
the Root Test is applied to the series
.
When
by the assumption and the notation that

implies that
therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When is fixed,
, and
with the assumption that and
the Root Test is applied to the series
.
When by the assumption and the notation that
implies that therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
Compare your answer with the correct one above
Determine whether the following statement is true or false:
Some power series such as
have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
Determine whether the following statement is true or false:
Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When
is fixed,
, and 
with the assumption that
and
the Root Test is applied to the series
.
When
by the assumption and the notation that

implies that
therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.
The proof of one case is as follows:
When is fixed,
, and
with the assumption that and
the Root Test is applied to the series
.
When by the assumption and the notation that
implies that therefore by the Root Test,
does not converge for any
and thus the radius of convergence of
is
.
Compare your answer with the correct one above