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alexmahone
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Give an example of a convergent series $\sum a_n$ for which $(a_n)^{1/n}\to 1$.
PS: I think I got it: $\sum\frac{1}{n^2}$
PS: I think I got it: $\sum\frac{1}{n^2}$
Alexmahone said:Give an example of a convergent series $\sum a_n$ for which $(a_n)^{1/n}\to 1$.
PS: I think I got it: $\sum\frac{1}{n^2}$
A convergent series is a sequence of numbers where the terms get closer and closer to a fixed value as the number of terms increases. In other words, the sum of the terms in the series approaches a finite limit.
One example of a convergent series is the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which has a sum of 2. This means that as you add more terms to the series, the sum will get closer and closer to 2.
A series is convergent if the limit of its terms approaches a finite number. This can be determined by using various tests such as the ratio test, comparison test, or integral test.
A convergent series has a finite sum, while a divergent series has an infinite sum. In other words, the terms in a convergent series approach a fixed value, while the terms in a divergent series do not have a limit.
Convergent series play an important role in calculus, as they allow us to use infinite sums to approximate values of functions. They also have applications in fields such as physics, engineering, and economics.