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    

If    

1. Compare to a harmonic series

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/b_n, which does seem to converge to 1. We can verify this: The limit comparison test says that in this case, both converge or both diverge. Since we know that the harmonic series diverges, A must also diverge.

2. Compare to a geometric series

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.

Related 'Sequences and Series' topics

• 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

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.