Solving the Limit of an Infinite Series

In summary, the conversation was about solving a limit problem using various methods. The solution was provided using the Riemann sum method, and it was suggested that the person who posted the problem demonstrate their own technique if they had one.
  • #1
jeffer vitola
26
0
hello ... I propose this exercise for you to solve on various methods ...\[\lim_{n \to{+}\infty}{\frac{1}{n}\sum_{i=1}^n({1+\frac{i}{n}}})^{-2}\]thanks

att
jefferson alexander vitola(Smile)
 
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  • #2
I have moved this topic, as it seems to be posted as a challenge rather than for help.
 
  • #3
jeffer vitola said:
\[\lim_{n \to{+}\infty}{\frac{1}{n}\sum_{i=1}^n({1+\frac{i}{n}}})^{-2}\]

This can be solved directly using the Riemann sum

\(\displaystyle \lim_{n \to{+}\infty} \sum_{i=1}^n \frac{n}{(n+i)^2}=\int^2_1 \frac{1}{x^2}\, dx = \frac{1}{2}\)
 
  • #4
ZaidAlyafey said:
This can be solved directly using the Riemann sum

\(\displaystyle \lim_{n \to{+}\infty} \sum_{i=1}^n \frac{n}{(n+i)^2}=\int^2_1 \frac{1}{x^2}\, dx = \frac{1}{2}\)

hello...interesting, but as I said in my previous forum topic or main focus is that you develop by various methods ... one can be evaluated by the summation properties and then calculate its limit for example,,,,,,,, if you are can't make exercise , there is not problem,,,, thanks...

att
jefferson alexander vitola (Smile)
 
  • #5
jeffer vitola said:
hello...interesting, but as I said in my previous forum topic or main focus is that you develop by various methods ... one can be evaluated by the summation properties and then calculate its limit for example,,,,,,,, if you are can't make exercise , there is not problem,,,, thanks...

att
jefferson alexander vitola (Smile)

Zaid has shown you a very straightforward method to evaluate the sum. Your initial post said only "solve on various methods." And this is what Zaid has done.

Why don't you demonstrate the technique you have? We expect that when people post problems as a challenge, they have a solution which they post if the problem has not been solved within about a week's time. Although this problem has been solved, but seemingly not to your satisfaction, it is now time for you to show us your solution.
 

FAQ: Solving the Limit of an Infinite Series

What is an infinite series?

An infinite series is a sum of an infinite number of terms, with a specific pattern or rule that determines the values of each term.

What is the limit of an infinite series?

The limit of an infinite series is the value that the series approaches as the number of terms increases to infinity.

How do you solve the limit of an infinite series?

To solve the limit of an infinite series, you can use a variety of mathematical techniques such as the ratio test, comparison test, or the integral test. These methods help determine whether the series converges or diverges, and if it converges, what value it approaches.

Why is solving the limit of an infinite series important?

Solving the limit of an infinite series is important because it allows us to determine the behavior of a series as the number of terms increases. This is useful in many areas of mathematics, including calculus, differential equations, and probability.

What are some real-world applications of solving the limit of an infinite series?

Solving the limit of an infinite series has many real-world applications, such as calculating the value of an investment over time, determining the maximum load a bridge can withstand, and predicting the weather patterns over a long period of time.

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