Matching each series to its test

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

Which test settles the series with terms 1 over n cubed the fastest?
- p-series with p equals 3
- Ratio test
- Root test
A ratio limit of 1 proves that a series converges.
Circle one: True False
Terms 1 over n times square root of n form a p-series. Type p.
Answer: ______________
Which test settles the series with terms 1 over n cubed the fastest?
- p-series with p = 3
- Ratio test
- Root test
- Alternating series test
Which test settles the series with terms n factorial over n to the n the fastest?
- p-series
- Integral test
- Ratio test
Why does the series with terms 1 over (n squared + 5) converge?
- Its terms sit below 1 over n squared, a convergent p-series
- Its terms sit above 1 over n, a divergent series
- The ratio test gives a limit of 2
- Its nth term tends to 5
Which test settles the series with terms n factorial over n to the n the fastest?
- Ratio test
- p-series
- Integral test
- nth term test
Why does the series with terms 1 over n squared plus 5 converge?
- The ratio test gives a limit below 1
- Its terms sit below 1 over n squared, a convergent p-series
- Its nth term tends to 5
The ratio test gives a limit of 1 for the harmonic series. What follows?
- Nothing: the test is inconclusive when the limit is 1
- The series converges
- The series diverges by the ratio test
For the series with terms n squared over 2 to the n, which test settles it and what happens?
- nth term test; it diverges
- Ratio test; it converges
- p-series with p equals 2; it converges

Answer key

For grown-ups. Fold this page away before handing over the rest.

p-series with p equals 3 · It is a p-series with p above 1, so convergence is immediate.
False · A limit of 1 is the silent case, so no conclusion follows.
1.5 · n times square root of n is n to the 1.5, above the cut off.
p-series with p = 3 · It is a p-series with p above 1, so convergence is immediate, while ratio and root both return the silent limit 1.
Ratio test · Factorials against exponentials are ratio test territory.
Its terms sit below 1 over n squared, a convergent p-series · Zero is below the terms and 1 over n squared is above them, so direct comparison with the convergent p-series wins.
Ratio test · Factorials against exponentials are ratio test territory, and the limit is 1 over e, below 1.
Its terms sit below 1 over n squared, a convergent p-series · Direct comparison with the convergent p-series transfers convergence.
Nothing: the test is inconclusive when the limit is 1 · A ratio limit of 1 is the known blind spot of the test.
Ratio test; it converges · The exponential in the denominator makes ratio the matching test.