Quick Q: Understanding (k+1)^2 in Series Calculations

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In summary, the highlighted part in the equation should be (2k+1)^2 instead of (k+1)^2. The third to last term is shown as k^2, but using the same method it should be 4k^2. There may be a mistake or something missing, but it explains why it is (2k+1)^2.
  • #1
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Should the highlighted part not be (k+1)^2, could anyone explain?>
 
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  • #2
Well, it makes sense to me that it would be (2k+1)^2. Since you're summing up to 2k+2, going backwards from the final term you would have (2k+2)^2 then (2k+2 - 1)^2 then (2k+2 - 2)^2, etc. which would make the second to last term (2k+1)^2 and not (k+1)^2. What doesn't make sense to me is that it shows the third to last term as k^2. Using the same method I just described, which I don't see a problem with, you get that the third to last term should be (2k+2 - 2)^2 = (2k)^2 = 4k^2. So, either there's a mistake or I'm missing something. If my reasoning is correct, though, it explains why it's (2k+1)^2
 
  • #3
elvishatcher said:
Well, it makes sense to me that it would be (2k+1)^2. Since you're summing up to 2k+2, going backwards from the final term you would have (2k+2)^2 then (2k+2 - 1)^2 then (2k+2 - 2)^2, etc. which would make the second to last term (2k+1)^2 and not (k+1)^2. What doesn't make sense to me is that it shows the third to last term as k^2. Using the same method I just described, which I don't see a problem with, you get that the third to last term should be (2k+2 - 2)^2 = (2k)^2 = 4k^2. So, either there's a mistake or I'm missing something. If my reasoning is correct, though, it explains why it's (2k+1)^2

That makes sense, thanks.
 
  • #4
Glad I could help - if you ever figure out some reason why I'm wrong and it should be k^2 not 4k^2, let me know
 

1. What is the purpose of understanding (k+1)^2 in series calculations?

The purpose of understanding (k+1)^2 in series calculations is to simplify and accurately calculate complex mathematical series. It allows for a more efficient and organized approach to solving problems involving series.

2. How is (k+1)^2 used in series calculations?

(k+1)^2 is used as a way to represent the terms in a series, where k is the current term and (k+1) is the next term. This allows for a systematic approach to calculating the sum of a series.

3. What is the significance of (k+1)^2 in series calculations?

The significance of (k+1)^2 is that it represents the relationship between consecutive terms in a series. It helps to simplify complex series by breaking them down into smaller, more manageable parts.

4. How does understanding (k+1)^2 in series calculations help in real-world applications?

Understanding (k+1)^2 in series calculations can help in real-world applications by providing a more accurate and efficient way to calculate complex series. It can also be applied in fields such as finance, engineering, and physics to solve problems involving series.

5. What are some common mistakes when using (k+1)^2 in series calculations?

Some common mistakes when using (k+1)^2 in series calculations include forgetting to include the +1 or using the wrong exponent. It is also important to carefully consider the starting value of k and the number of terms in the series to avoid errors in calculations.

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