The **limit comparison test** is similar to the comparison test in that you use another series to show the convergence or divergence of a desired series. Suppose we have two series

and
where *a*_{n} >0 and *b*_{n }> 0.

If
(i.e., if the ratio of the terms tends to a finite number as *n* goes to infinity), then both series converge or both series diverge. By picking a suitable *B*, usually a *p*-series, we can use this test to determine whether or not *A* converges.

## 1. Compare to a harmonic series

## 2. Compare to a geometric series

## Other 'Sequences and Series' topics

See About the calculus applets for operating instructions. |

The applet shows the series
A useful way to pick a comparison series when the target series uses a rational expression is to divide the highest power of *n* in the numerator by the highest power of *n* in the denominator, which in this case yields
The table shows the ratio *a _{n}*/

Select the second example from the drop down menu, showing
Use the same guidelines as before, but include the exponential term also:
The limit of the ratio seems to converge to 1 (the "undefined" in the table is due to the *b* terms getting so small that the algorithm thinks it is dividing by 0), which we can verify:
The limit comparison test says that in this case, both converge or both diverge. Since
the *B* series is a geometric series with *r* = 1/2, which we know converges, so* A* also converges.

- Sequences
- Series
- Integral Test
- Comparison Test
- Limit Comparison Test
- Ratio Test
- Alternating Series and Absolute Convergence
- Power Series & Interval of Convergence
- Taylor Series & Polynomials
- Lagrange Remainder

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