Technology
Calculating the Sum of an Infinite Series: Understanding the Method
Calculating the Sum of an Infinite Series: Understanding the Method
In this article, we will delve into the fascinating world of infinite series and explore how to calculate the sum of a specific series. We will walk through the process step-by-step, breaking down the mathematics involved to ensure a deep understanding of the topic.
The Series in Focus
The series we are examining is given by:
S (sum_{n1}^{infty} frac{n}{2n-1!})
Understanding the General Term
The general term of the series is:
a_n (frac{n}{2n-1!})
Methodology for Summing the Series
Step 1: Simplifying the General Term
First, let's simplify the general term by rewriting the series:
S (sum_{n1}^{infty} frac{n}{2n-1!} frac{1}{2} sum_{n1}^{infty} frac{2n-1}{2n-1!})
Step 2: Manipulating the Series
Now, we can manipulate the series further:
2S (sum_{n1}^{infty} frac{2n}{2n-1!} - sum_{n1}^{infty} frac{1}{2n-1!})
Simplifying the expression, we get:
2S (sum_{n1}^{infty} frac{1}{2(n-frac{1}{2})!} - sum_{n1}^{infty} frac{1}{2n-1!})
Step 3: Splitting the Series into Two Parts
We now split the series into two parts:
2S (sum_{n1}^{infty} frac{1}{2n!} - sum_{n1}^{infty} frac{1}{(2n-1)!})
Using Taylor Series Expansion
Next, we will use the Taylor series expansion of the exponential function, e^x,to help us solve the series:
e^x 1 (frac{x}{1!} frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} ...)
Evaluating the Series at Specific Values
First, we evaluate the series at x1,and x-1,to get:
e 1 (frac{1}{1!} frac{1}{2!} frac{1}{3!} frac{1}{4!} ...)
e^{-1} 1 - (frac{1}{1!} frac{1}{2!} - frac{1}{3!} frac{1}{4!} - ...)
Deriving the Series for S_1,and S_2,
Next, we multiply e,and e^{-1},and then divide by 2 to get the series for S_1,and S_2,respectively:
(frac{e e^{-1}}{2} 1 frac{1}{2!} frac{1}{4!} frac{1}{6!} ...)
(frac{e - e^{-1}}{2} frac{1}{1!} frac{1}{3!} frac{1}{5!} ...)
Final Step: Finding the Sum of the Original Series
Now, using the results for S_1,and S_2,we can find the sum of the original series:
2S S_1 - S_2 (frac{ee^{-1} - 2}{2}) - (frac{e - e^{-1} - 2}{2}) (frac{e^{-1}}{2})
Therefore, the sum of the series is:
S (frac{e^{-1}}{2})
Conclusion
In this article, we have successfully determined the sum of the given infinite series using advanced mathematical techniques involving series expansion and manipulation. This method not only provides a clear understanding of the concept but also showcases the power of—and beauty in—mathematical problem-solving.