Series Exercises

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

As n approaches infinity, n! will overpower (n + 1) so

The series might converge, but we don't know for sure. The statement

" converges"

can't be decided using only the Divergence Test.

Answer

For the series, look at the limit of the terms. If

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

As n approaches infinity, the exponential 2ⁿ overpowers n². This means

so the sequence diverges. The statement

" diverges"

is true.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

Since 0.9 is less than 1,

This means the statement

" might converge"

is true.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

Since 1.1 is greater than 1,

This means we know the series diverges, so the statement

" converges"

is false.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

The limit of the terms is

so the series may or may not converge. Since we're only using the Divergence Test, we can't tell whether the statement

" diverges"

is true or not.

Answer

The terms converge to 0, so this series might converge. Further investigation is needed.

Answer

This series diverges since the limit of the terms, , diverges.

Answer

The limit of the terms is . Since this limit is not equal to 0, the series diverges.

Answer

The nth term of this series simplifies to . Since , this sequence might converge.

Answer

If we write out the nth term of this series we get

The n and the 2 cancel from the numerator and the denominator. This leaves us with an expression with nothing in the denominator:

(n – 1) · (n – 2) · ...3

As n approaches ∞, the terms given by this expression will also approach infinity. Thus the series diverges.