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United States Calculus 2 - Infinite Series

  • Thread starter GreenPrint
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  • #1
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Homework Statement



Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given.

ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k

Homework Equations





3. The Attempt at a Solution [/b

| ln(128) - 7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000
subtracted ln(128) from both sides
|-7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000 - ln(128)
simplified the negative on the left hand side and absolute value
7*sum[k=1,n] of 1/k < 1/10,000 - ln(128)
divided through by 7
sum[k=1,n] of 1/k < 1/70,000 - ln(128)/7

I'm unsure were to go from here, thank you for any help you can provide me.
 

Answers and Replies

  • #2
33,265
4,965

Homework Statement



Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given.

ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k

Homework Equations





3. The Attempt at a Solution [/b

| ln(128) - 7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000
subtracted ln(128) from both sides
|-7*sum[k=1,n] of (-1)^(k+1)/k | < 1/10,000 - ln(128)
simplified the negative on the left hand side and absolute value
7*sum[k=1,n] of 1/k < 1/10,000 - ln(128)
divided through by 7
sum[k=1,n] of 1/k < 1/70,000 - ln(128)/7

I'm unsure were to go from here, thank you for any help you can provide me.

There's a remainder theorem for alternating series. Take a look at it.
 

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