In a recent video, the Numberphile team attempted to prove their answer to the question, “How do you evaluate the sum of n2 from n1 to infinity?” by using the Euler zeta function, which is commonly used in physics. The Euler zeta function was named after the prolific 18th-century mathematician Leonhard Euler.
convergent or divergent
A series can be either convergent or divergent. If the sum of n2n is convergent, then the series has a minimum error value (mean) of N plus one. If the sum is divergent, then the error value is 0.001.
nth term test
A series is a collection of numbers. Its nth term is known as its index. As the number n approaches infinity, the sequence is said to be converging. Alternatively, a series may be diverging if its nth term is higher than its nth term. An alternating series test is helpful if the series has multiple negative terms. A direct comparison test is also useful. Other tests are often easier.
If the series is divergent, the nth term test indicates that it is not converging. This is because the series does not settle down. It must never be zero at infinity. It is a simple test. Moreover, it is easy to apply. Once you have defined the value of n, you can determine whether the series is converging or diverging.
The nth term test is the most popular test to evaluate the sum of n2 from n1 to infinity. Its conclusion is the same as the Ratio Test, but it can diverge as well. A test of this kind can be useful when the number of terms gets small or large too fast. If it is too small, it would signal a series divergence.