Solve Converging Sequence: 1 + 1/8 + 1/27 + 1/64...

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SUMMARY

The series 1 + 1/8 + 1/27 + 1/64 is an infinite series represented by the formula 1/n^3, which converges as n approaches infinity. The convergence can be confirmed using the Integral Test, which shows that the limit of 1/x^6 is finite (0). While there is no simple closed form for the sum of this series, it is related to the Riemann zeta function ζ(3). For practical purposes, a decimal approximation is sufficient for high school calculus classes.

PREREQUISITES
  • Understanding of infinite series and convergence
  • Familiarity with the Integral Test for convergence
  • Knowledge of the Riemann zeta function ζ(s)
  • Basic calculus concepts, including limits
NEXT STEPS
  • Study the Integral Test for convergence in detail
  • Learn about the Riemann zeta function ζ(3) and its applications
  • Explore numerical methods for approximating series sums
  • Investigate other convergence tests such as the Comparison Test and Ratio Test
USEFUL FOR

Students in high school calculus, mathematics educators, and anyone interested in understanding infinite series and their convergence properties.

Hockeystar
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Homework Statement


Solve 1 + 1/8 + 1/27 + 1/64...


Homework Equations


Stuck here.


The Attempt at a Solution



I know the series is infinitetly adding 1/n^3. I know it should converge. How do I solve as lim x--> infinity. Teacher never gave us any info, he just said use number sense.
 
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Hockeystar said:

Homework Statement


Solve 1 + 1/8 + 1/27 + 1/64...


Homework Equations


Stuck here.


The Attempt at a Solution



I know the series is infinitetly adding 1/n^3. I know it should converge. How do I solve as lim x--> infinity. Teacher never gave us any info, he just said use number sense.
What exactly does "solve" mean? Are you supposed to find the sum of this series, or do you only need to say that it converges or diverges?

If it's the latter, do you know any tests you can use to determine convergence or divergence?
 
try integral test
 
My teacher wanted to know the sum. Integral test is as n--> infinity, 1/x^6 is finite (0). Therefore the series converges. How do I find the sum then? Sohuld I just try trial and error and see what it seems to approach?
 
I don't think there's a good closed form for that sum other than ζ(3), the Riemann zeta function.
 
I could do 1/n^2 infinite sum is (pi^2)/6 and then multiply by infinite 1/n but that series diverges. Found solution on wikipedia. http://en.wikipedia.org/wiki/Apéry's_constant

I think my teacher was just looking for a decimal approx (only H.S calc. class) Thanks for the help everyone.
 

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