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if-the-series-n-1-1-n-2-converges-to-k-find-the-convergence-value-of-n-1-1-2n-1-2-




Question Number 209837 by lmcp1203 last updated on 23/Jul/24
  if the series Σ_(n=1) ^∞ (1/n^2 ) converges to k .  find  the convergence value of Σ_(n=1) ^∞ (1/((2n+1)^2 ))
iftheseriesn=11n2convergestok.findtheconvergencevalueofn=11(2n+1)2
Answered by mr W last updated on 23/Jul/24
Σ_(n=1) ^∞ (1/n^2 )=k  Σ_(n=1) ^∞ (1/((2n)^2 ))=(k/4)  Σ_(n=1) ^∞ (1/((2n)^2 ))+Σ_(n=1) ^∞ (1/((2n+1)^2 ))+(1/1^2 )=Σ_(n=1) ^∞ (1/n^2 )=k  ((3k)/4)+Σ_(n=1) ^∞ (1/((2n+1)^2 ))+1=k  ⇒Σ_(n=1) ^∞ (1/((2n+1)^2 ))=((3k)/4)−1 ✓
n=11n2=kn=11(2n)2=k4n=11(2n)2+n=11(2n+1)2+112=n=11n2=k3k4+n=11(2n+1)2+1=kn=11(2n+1)2=3k41
Answered by lmcp1203 last updated on 23/Jul/24
thanks
thanks

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