Sum of infinite series - 1/n^2

In summary, the sum of the infinite series 1/n^2 is approximately 1.64493406685, also known as the Basel problem or Basel sum. The sum is calculated using techniques such as convergence tests and limit theorems, by taking the limit as n approaches infinity of the partial sums. This value has real-life applications in physics, engineering, mathematical models and computer algorithms. The sum of an infinite series can be negative, but for the series 1/n^2 it is always positive. Not all infinite series have a finite sum, as it depends on the series and its convergence conditions.
  • #1
praharmitra
311
1
How do you go about finding the sum, [tex]\sum \frac{1}{n^2}[/tex].

I remember studying it earlier, but don't quite remember how it was done..just tell me the method. i'll figure the rest out.
 
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1. What is the sum of the infinite series 1/n^2?

The sum of the infinite series 1/n^2 is approximately 1.64493406685. This value is known as the Basel problem or the Basel sum, named after the Swiss mathematician who first solved it in the 18th century.

2. How is the sum of an infinite series calculated?

The sum of an infinite series is calculated using mathematical techniques such as convergence tests and limit theorems. For the series 1/n^2, the sum is found by taking the limit as n approaches infinity of the partial sums.

3. What real-life applications does the sum of an infinite series have?

The sum of an infinite series has many real-life applications, particularly in physics and engineering. For example, it is used in calculating the electrical capacitance of a parallel plate capacitor and the energy stored in an inductor. It is also used in various mathematical models and computer algorithms.

4. Can the sum of an infinite series be negative?

Yes, the sum of an infinite series can be negative. For example, the series -1/2^n has a sum of -1, as shown by the geometric series formula. However, for the series 1/n^2, the sum is always positive.

5. Is the sum of an infinite series always a finite value?

No, the sum of an infinite series may not always be a finite value. It depends on the series and whether it satisfies certain conditions for convergence. For example, the series 1/n diverges and does not have a finite sum, while the series 1/n^2 converges and has a finite sum.

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