nth Term Test
Definition
The nth Term Test is a method used to determine whether an infinite series converges or diverges by examining the behavior of its individual terms. If the limit of the sequence of terms as n approaches infinity is not zero, then the series diverges. If the limit is zero, further tests are needed to determine convergence.
Analogy
Imagine you have a jar filled with marbles. You want to know if you can keep adding more and more marbles without it overflowing. The nth Term Test helps you analyze each marble individually to see if they get smaller and smaller (converge) or stay relatively large (diverge).
Related terms
Converges: When an infinite series converges, it means that the sum of all its terms approaches a finite value as more terms are added.
Diverges: When an infinite series diverges, it means that the sum of its terms does not approach a finite value but instead grows infinitely larger or oscillates between different values.
Geometric Series: A geometric series is a specific type of infinite series where each term is obtained by multiplying the previous term by a constant ratio.
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Study guides (1)
Practice Questions (11)
Apply the nth term test to determine whether the series $\sum \frac{1}{n}$ converges or diverges.
What does the nth term test state?
Apply the nth term test to determine whether the series $\sum \frac{1}{n^2}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{1}{\sqrt{n}}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{1 + n^5}{6n^5}$ converges or diverges.
What is the nth term test also called?
Apply the nth term test to determine whether the series $\sum \frac{n+7}{2n}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{3n^6}{n^6+9}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{1}{n!}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{(-3)^n}{\sqrt{n}}$ converges or diverges.
Apply the nth term test to determine whether the series $\sum \frac{4n^2-n^5}{5+3n^5}$ converges or diverges.
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