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Divergence and Integral Tests

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Objectives

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Divergence Test

  • The divergence test allows us to determine the divergence of a series.

Theorem
Divergence Test

If or does not exist, then the series diverges.

Proof
Proving the Divergence Test

  • Consider the series .
    • If this series is convergent, then the sequence of partial sums converges.
    • And so:
  • Therefore, if converges, as , .
  • But what if ? Then, the series diverges.
  • Despite the usefulness of this theorem, there is one important caveat.

Note
Limitations of the Divergence Test

It is important to know that if

we cannot make any conclusion about the convergence of the series .
Therefore, although the divergence test can show if a series diverges, it cannot be used to prove that a series converges.

  • One example of this limitation is the harmonic series.
    • We can say that .
    • However, the harmonic series diverges.

Integral Test

  • One application of the integral in series is that it can be used to prove that a series converges or diverges.

Theorem
Integral Test

Let be a series where for all .
Suppose that there exists a function and a positive integer where:

  • is continuous,
  • is decreasing
  • for all integers .

Then:

both diverge or both converge.

Proof
Proving the Integral Test

Let be a series where for all .
Suppose that there exists a continuous, positive, decreasing function where for all positive integers .

Then, for any positive integer, the partial sum satisfies.

Therefore, if converges, then the sequence of partial sums is bounded.
Since is an increasing sequence, and a bounded sequence at the same time, then by Monotone Convergence Theorem, it converges.
Therefore, we can finally say that:

Likewise, if for any integer , the partial sum satisfies

If , then is an unbounded sequence, and therefore diverges.

As a result, the series diverges.

We conclude that:

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