Convergence of a Series: Is My Approximation Accurate Enough?

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The discussion centers on the convergence of a series, specifically the approximation of the sum S6 in relation to S5. The user calculated S5 = -0.28347 and S6 = -0.28347, concluding that Sn is approximately -0.2835. The consensus among participants confirms that reaching the same digits in the approximation indicates sufficient accuracy, validating the user's conclusion that further terms are unnecessary.

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ardentmed
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Hey guys,

I just wanted to run a quick series question by you guys just to confirm my answer. I'm doubting whether or not I should keep going or if S6 is enough.

View attachment 2804

I got S5 = -0.28347 and S6 = -0.28347, so that is where I concluded than Sn ~ -0.2835.

I would appreciate it if someone could look over this for me and tell me how I did.

Thanks in advance.
 

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ardentmed said:
Hey guys,

I just wanted to run a quick series question by you guys just to confirm my answer. I'm doubting whether or not I should keep going or if S6 is enough.

https://www.physicsforums.com/attachments/2804

I got S5 = -0.28347 and S6 = -0.28347, so that is where I concluded than Sn ~ -0.2835.

I would appreciate it if someone could look over this for me and tell me how I did.

Thanks in advance.

When you get the sum giving you the same digits up to your level of accuracy when you add more terms, then you have reached your approximation. So yes, what you have done is enough.
 

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