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Wize University Calculus 2 Textbook > Sequences and Series

Integral Test

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Integral Test

Consider the series ∑n=N∞an\sum\limits^\infty_{n=N}a_n, where ana_n is a sequence of positive terms.
Let f(n)=anf(n)=a_n and assume that ff is continuous, positive and decreasing.
Then

1. ∫N∞f(x)  ⁣dx{\displaystyle\int}^\infty_Nf(x)\de{x} converges if and only if ∑n=N∞an\sum\limits^\infty_{n=N}a_n converges.

2. ∫N∞f(x)  ⁣dx{\displaystyle\int}^\infty_Nf(x)\de{x} diverges if and only if ∑n=N∞an\sum\limits^\infty_{n=N}a_n diverges.

Identifying Clues

  • The terms ana_n look integrable
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.

Practice Question

Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.

Extra Practice
Practice: Integral Test
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.
Practice: Integral Test
Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n} converges or diverges.
Practice: Integral Test
Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}} converges or diverges.