Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?

In summary, the conversation discusses the possibility of converting a given limit into an integral or a more elegant form as N tends to infinity. It is mentioned that the series converges quickly when plotted or evaluated, and the value of N can be changed using a slider. The limit is then evaluated for positive values of x, but it is noted that the summation must also be taken into account. The limit is shown to exist and be non-zero, as demonstrated by the graph.
  • #1
SatyaDas
22
0
I wonder if the limit of the following can be converted into integral or some elegant form as N tends to infinity:
\[ \sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right) \]

If we plot or evaluate the value then it does appear that the series converges very fast.

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  • #2
I forgot to mention that the above graph is plotted as function of x and value of N can be changed by using the slider and that shows that the graph stabilizes pretty fast if N is increased.
 
  • #3
\[ \lim_{N\rightarrow\infty}\frac{x}{2^n}\sin^2\left(\frac{x}{2^n}\right)\rightarrow0\times0^2\rightarrow0 \] but as the limit is taken over positive $x$ the limit tends to infinity.
 
  • #4
Greg said:
\[ \lim_{N\rightarrow\infty}\frac{x}{2^n}\sin^2\left(\frac{x}{2^n}\right)\rightarrow0\times0^2\rightarrow0 \] but as the limit is taken over positive $x$ the limit tends to infinity.
You missed taking the summation into account. The lower case 'n' is the index for summation and the expression is summed till n=N. We need to find the limit of the sum as the upper case 'N' tends to infinity.
And certainly the limit exists and is non zero that is demonstrated by the graph also. You can you the slider in the graph to change the value of N and see that the graph stabilizes pretty fast.
 
Last edited:

Related to Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?

1. What does "Summing Series Limit as N→∞" mean?

The notation "N→∞" represents the limit as the number of terms in a series approaches infinity. The summing series limit refers to finding the value that a series approaches as the number of terms increases without bound.

2. Why is finding the summing series limit as N→∞ important?

Finding the summing series limit is important in various fields of science, such as physics, engineering, and economics. It allows us to make predictions and analyze systems that involve infinite sequences or continuous processes.

3. How do you calculate the summing series limit as N→∞?

The summing series limit as N→∞ is calculated using various mathematical techniques, such as the ratio test, the integral test, and the comparison test. These methods help determine whether a series converges or diverges, and if it converges, what value it approaches.

4. What is the difference between a convergent and a divergent series?

A convergent series is one that has a finite sum, meaning that the value of the series approaches a specific number as the number of terms increases. In contrast, a divergent series does not have a finite sum, and the value of the series either approaches infinity or oscillates between different values as the number of terms increases.

5. Can the summing series limit as N→∞ be applied to real-world problems?

Yes, the concept of summing series limit as N→∞ has various real-world applications, such as calculating the total distance traveled by an object in continuous motion, determining the value of investments with continuous compounding, and analyzing the stability of systems in physics and engineering.

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