# Video: Using the πth Term Divergence Test

Use the πth term divergence test to determine whether the series β_(π = 1) ^(β) (3^(π)/5^(π)) is divergent or whether the test is inconclusive.

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### Video Transcript

Use the πth term divergence test to determine whether the series the sum from π equals one to infinity of three π over five π is divergent or whether the test is inconclusive.

Letβs begin by recalling what we actually know about the πth term divergence test. This tells us that if the limit as π approaches infinity of π sub π does not exist, or if itβs not equal to zero, then the series the sum from π equals one to infinity of π sub π is divergent. And now whilst at series that converges weβll have a limit that is equal to zero. If weβre just working with the limit, and thatβs equal to zero, we canβt actually tell whether the series converges or diverges. And we say that the test fails or itβs inconclusive.

And so we let π sub π be equal to three to the πth power over five to the πth power. And so we want to find the limit as π approaches infinity of three to the πth power over five to the πth power. Now if we were to substitute in π equals infinity, we just end up with infinity over infinity, which we know to be undefined. And so weβre going to manipulate our expression somewhat. Weβre going to write it as the limit as π approaches infinity of three-fifths to the πth power.

Now this is really useful because three-fifths is a fraction which is less than one. And so if we have π is equal to two, we get three-fifths all squared, which is nine over 25. Thatβs less than three-fifths. If π is equal to three, we have three-fifths cubed, which is equal to 27 over 125, which is less than nine over 25.

Each value of π makes the fraction three-fifths to the πth power ever smaller. And that actually means that the limit as π approaches infinity of three-fifths to the πth power must be zero. We recall that this doesnβt actually tell us anything about whether the series converges or diverges. It fails or itβs inconclusive.