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- Homework Statement
- Calculate ##\sin 10## using the sine series, correct upto two places of decimal.

- Relevant Equations
- The sine series

The above is one specific example of this type of problem that we often encounter in our course. The series can be, in general, anything like the log series, cosine series, etc. We are supposed to solve this problem in Python, but that doesn't matter here.

When this question was put forward for the first time in class, I proposed that if the difference between the sum upto ##(n+1)## terms and the sum upto ##n## terms is less ##10^{-x}##, then we have obtained the sum correct to ##x## decimal places. Basically the condition boils down to this: $$S_{n+1} - S_n \ = \ t_{n + 1} \ < \ 10^{-x},$$where ##t_{n + 1}## is the ##(n+1)##th term. The Professor said that this is correct, and applicable to any series.

I stop calculating the series at ##t_{n + 1}## when the above condition is met. But is it possible that somewhere down the line, after adding many more terms, a correction suddenly pops up in the ##x##th decimal place? Say for some series, the ##n##th term is##<10^{-2}##, and as per the question, I stop calculating the sum further. But if I had calculated for many more terms, I would have reached a state where the sum would have been ##2.33995223##, and after adding the next term (which is of the order of ##10^{-3}##), it becomes ##2.346995##. Is this possible? If yes, then the condition put above is wrong, isn't it?

When this question was put forward for the first time in class, I proposed that if the difference between the sum upto ##(n+1)## terms and the sum upto ##n## terms is less ##10^{-x}##, then we have obtained the sum correct to ##x## decimal places. Basically the condition boils down to this: $$S_{n+1} - S_n \ = \ t_{n + 1} \ < \ 10^{-x},$$where ##t_{n + 1}## is the ##(n+1)##th term. The Professor said that this is correct, and applicable to any series.

I stop calculating the series at ##t_{n + 1}## when the above condition is met. But is it possible that somewhere down the line, after adding many more terms, a correction suddenly pops up in the ##x##th decimal place? Say for some series, the ##n##th term is##<10^{-2}##, and as per the question, I stop calculating the sum further. But if I had calculated for many more terms, I would have reached a state where the sum would have been ##2.33995223##, and after adding the next term (which is of the order of ##10^{-3}##), it becomes ##2.346995##. Is this possible? If yes, then the condition put above is wrong, isn't it?