Infinite Series Doubt: Last Step Explained

In summary, the conversation is about understanding the last step of the first sum on a provided attachment. The formula for the sum of a finite geometric series is being used, where the value of the infinite series is obtained by taking the limit of the sum of the first n terms. The person asking the question is unsure about the concept and apologizes for their weak basic understanding.
  • #1
RoughRoad
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0
On the provided attachment, I have problem understanding the last step of the 1st sum of that page. Can anyone explain to me what is being done after the second-last step?
 

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  • #2
They're using the formula for the sum of a finite geometric series:
[tex]\sum_{k=1}^{n}r^k = r\frac{1-r^n}{1-r}[/tex]
 
  • #3
How is it a finite series? Aren't the term going till infinity? Sorry but I am actually a litle weak on my basics.
 
  • #4
The term s_n is the sum of just the first n terms, which is a finite series. The value of the infinite series is obtained by taking the limit of s_n as n→∞.
 
  • #5
Thanks! :-)
 

1. What is an infinite series?

An infinite series is a mathematical concept that represents the sum of an infinite number of terms. Each term in the series is added together, starting from the first term, to determine the total sum.

2. What is the "last step" in an infinite series?

The "last step" in an infinite series refers to the final calculation or operation that is performed to determine the sum of the series. This could be adding the last term to the previous sum, multiplying the last term by a constant, or any other operation depending on the specific series.

3. Why is the last step important in understanding infinite series?

The last step is important because it is the final calculation that determines the total sum of the infinite series. It is often the most crucial step in solving the series and can greatly impact the final result.

4. How can the last step be explained?

The last step can be explained by breaking down the infinite series into smaller, more manageable steps. This can include identifying patterns in the terms, using mathematical formulas or rules, or applying known strategies for solving specific types of series.

5. Are there different methods for approaching the last step in an infinite series?

Yes, there are various methods for approaching the last step in an infinite series. These can include using algebraic manipulation, using geometric visualizations, or applying specific techniques such as telescoping or partial sums. The best method to use may depend on the specific series and the individual's own problem-solving skills.

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