Messing around with summation operators

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SUMMARY

The discussion centers on the manipulation of summation operators, specifically the series defined as sn = ∑k=1n ((-1)k+1)/k. Participants seek clarification on how to express s2n using summation notation. The attempts provided include s2n = ∑k=12n ((-1)2k+1)/2k and s2n = ∑k=12n ((-1)2k+2)/2k, both of which are incorrect. The discussion emphasizes the importance of understanding the properties of alternating series.

PREREQUISITES
  • Understanding of alternating series and their convergence
  • Familiarity with summation notation and index manipulation
  • Basic knowledge of mathematical proofs and series
  • Experience with mathematical notation and expressions
NEXT STEPS
  • Study the properties of alternating series and their convergence criteria
  • Learn about the manipulation of summation indices in mathematical proofs
  • Explore examples of series proofs involving alternating series
  • Practice writing summation expressions for various series
USEFUL FOR

Students in mathematics, particularly those studying calculus or series, educators teaching series manipulation, and anyone interested in enhancing their understanding of summation operators and alternating series.

gcamilo
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Homework Statement



I'm just not sure how to change the operators in summation, can anyone help?

Let [tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

what is [tex]s_{2n}[/tex]?

Homework Equations



[tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

The Attempt at a Solution



[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k[/tex]

or

[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+2})/2k[/tex]

This is in order to figure out a series proof, I just think I am really naive.
 
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gcamilo said:

Homework Statement



I'm just not sure how to change the operators in summation, can anyone help?

Let [tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

what is [tex]s_{2n}[/tex]?


Homework Equations



[tex]s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k[/tex]

The Attempt at a Solution



[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k[/tex]
No
gcamilo said:
or

[tex]s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+2})/2k[/tex]
No
gcamilo said:
This is in order to figure out a series proof, I just think I am really naive.

How would you write s5 as a summation (not expanded)? How about s10? Don't overthink this problem.
 
Thanks! I guess I tried to overthink, I've just seen people do amazing things with summation sings and sequences, I just tried to imitate.
 

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