The nth term test can only prove divergence: if the terms do not approach zero, the series diverges.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Geometric series
Converges exactly when the common ratio is smaller than 1 in absolute value.
p-series
The sum of 1/nᵖ converges exactly when p is greater than 1. At p = 1 it is the harmonic series, and it diverges.
The ratio test
Take the limit of the ratio of consecutive terms. Below 1 converges, above 1 diverges, and exactly 1 tells you nothing.
A sensible order of attack
Check the nth term test first. Recognise geometric or p-series. Then try comparison, ratio, alternating series or integral test as the form suggests. Having an order prevents flailing.
What the form signals
Factorials or nth powers suggest the ratio test. Rational expressions in n suggest comparison with a p-series. Alternating signs suggest the alternating series test. The shape of the terms points to the tool.
Every test has conditions
Comparison and integral tests require positive terms; the integral test also requires the function to be decreasing. Stating that the conditions hold is part of a complete justification and is graded.
Inconclusive is a real outcome
A ratio test giving exactly 1 tells you nothing and another test must be tried. Reporting an inconclusive test and moving on is correct; treating it as divergence is not.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Omar Pick a Test
Omar decides about the sum of 1/n².
- Step 1
The terms go to zero, so the nth term test is silent.
Step 4: Your Turn
Practice makes it stick.
A p-Series
Problem 1 of 2
Sum of 1/n³. Does it converge? 1 for yes, 0 for no.
The Borderline
Problem 2 of 2
Sum of 1/n. Does it converge? 1 for yes, 0 for no.
Converge or Diverge
1 of 8
Sum of 1/n⁴. Converge? 1 for yes, 0 for no.
2 of 8
Geometric with r = 0.9. Converge? 1 for yes, 0 for no.
3 of 8
Geometric with r = 1.1. Converge? 1 for yes, 0 for no.
4 of 8
Ratio test limit is 0.4. Converge? 1 for yes, 0 for no.
5 of 8
Ratio test limit is exactly 1. Is the test decisive? 1 for yes, 0 for no.
6 of 8
Sort each series by whether it converges.
Tap something to move it.
- Empty
- Empty
7 of 8
Terms approach 3, not 0. Converge? 1 for yes, 0 for no.
8 of 8
Match each series to the test that settles it most directly.
Tap a card on the left to start.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Sum of 1/n^0.5. Converge? 1 for yes, 0 for no.
What You Learned
- The nth term test can only prove divergence.
- A geometric series converges when |r| < 1; a p-series when p > 1.
- The ratio test is decisive except when its limit is exactly 1.