Practice question · Sort into groups
Sort each series by whether it converges.
Groups: Converges · Diverges
- The sum of 1/n, the harmonic series
- The sum of 2^n
- The sum of n/(n+1)
- The sum of 1/2^n
- The sum of 0.9^n
Hints
- For a geometric series check whether the ratio is smaller than 1 in size.
- One of these has terms shrinking to zero and diverges anyway.
Show the answer
Converges: The sum of 1/2^n, The sum of 0.9^n
Diverges: The sum of 1/n, the harmonic series, The sum of n/(n+1), The sum of 2^n
Why
Ratios of 1/2 and 0.9 both lie below 1, so those converge; a ratio of 2 blows up. The terms of n/(n+1) tend to 1 rather than 0, so it fails the term test outright. The harmonic series is the famous case: its terms do reach zero, yet it still diverges.
Practise Sequences and Series
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