P-SERIES TEST FOR CONVERGENCE AND DIVERGENCE

Consider the inifinite series given below.

If we want to check whether the above infinite series converges or diverges using p-series test, the nth of the series an should be in the form given below.

Then the given infinite series is in the form of

The above infinite series,

For each of the following infinite series, determine whether it converges or diverges.

Problem 1 :

Solution :

Since p = 4 > 1, the given series converges.

Problem 2 :

Solution :

Since p =  = 0.666..... < 1, the given series diverges.

Problem 3 :

Solution :

Since p = ⁵⁄₄ = 1.25 > 1, the given series converges.

Problem 4 :

Solution :

Since p =  = 0.2 < 1, the given series diverges.

Problem 5 :

Solution :

Since p = ³⁄₂ = 1.5 > 1, the given series converges.

Problem 6 :

Solution :

Since p =  = 0.666..... < 1, the given series diverges.

Problem 7 :

Solution :

Since p = ½ = 0.5 < 1, the given series diverges.

Problem 8 :

Solution :

Since p = -½ = -0.5 < 1, the given series diverges.

Problem 9 :

Solution :

Since p = π - e < 1, the given series diverges.

Problem 10 :

Solution :

Since p = 2e - π > 1, the given series converges.

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