Exercises relating to the Ratio and Root Tests.

Consider the infinite series Apply the Ratio Test: Apply the Root Test: Both tests
indicate that the series convergesdiverges
.

Oftentimes the Ratio Test is easier to apply than the Root Test when dealing with
factorials. Use the Ratio Test to determine convergence or divergence of the series In
the spaces below, record the eponymous “ratio” in the first blanks, simplify it in the
second blanks, and then record the limit. The test indicates convergenceindicates divergenceis inconclusive
.

Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

Apply the Ratio Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

Oftentimes the Root Test is easier to apply when terms have large exponents
(growing faster than a constant times ). Use the Root Test to determine
the convergence or divergence of the series First say what should quantity
should have its -th root taken, then simplify, and lastly record the value
of the limit. The test indicates convergenceindicates divergenceis
inconclusive
.

Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

Apply the Root Test to the series given below. The test indicates convergenceindicates divergenceis inconclusive
.

### Sample Quiz Questions

Determine which of the following three infinite series will lead to inconclusive results for the Ratio Test and then determine whether that series is convergent or divergent.

I inconclusive, converges I inconclusive, diverges II inconclusive, converges II
inconclusive, diverges III inconclusive, converges III inconclusive, diverges

The first series will give an inconclusive result for the Ratio Test because However,
we know that the harmonic series diverges and that so by direct comparison to the
harmonic series, series I must diverge.