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Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series converges.
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
can't be decided using only the Divergence Test.
The series diverges.
As n approaches infinity, the exponential 2n overpowers n2. This means
so the sequence diverges. The statement
The series might converge.
Since 0.9 is less than 1,
This means the statement
" might converge"
Since 1.1 is greater than 1,
This means we know the series diverges, so the statement
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
is true or not.
Use the divergence test to determine if the following series MUST diverge.
The terms converge to 0, so this series might converge. Further investigation is needed.
This series diverges since the limit of the terms, , diverges.
The limit of the terms is . Since this limit is not equal to 0, the series diverges.
The nth term of this series simplifies to . Since , this sequence might converge.
If we write out the nth term of this series we get
The n and 2 cancel from the numerator and the denominator. This leaves us with an expression that has 2 on the bottom and (n – 1) multiplied by a lot of stuff in the top:
As n approaches ∞, the terms given by this expression will also approach infinity. Thus the series diverges.