The number of terms needed to estimate the exact sum within .01 depends on the specific series being evaluated. In general, the more terms you include in the series, the closer your estimate will be to the exact sum. However, there are also mathematical techniques and formulas that can be used to determine the minimum number of terms needed for a given level of accuracy.
One example is the series for pi, which is an infinite series that converges very slowly. It requires a large number of terms (over 10,000) to estimate the exact sum within .01.
Yes, there are techniques like error estimation and convergence acceleration that can be used to improve the accuracy of the estimate without adding more terms to the series. These methods involve manipulating the terms of the series in a specific way to reduce the error in the final result.
Yes, there are series that converge quickly and only require a small number of terms to estimate the exact sum within .01. One example is the series for the natural logarithm, which converges rapidly and only needs a few terms for a very accurate estimate.
The accuracy of the estimate for a given number of terms can be determined by calculating the error, which is the difference between the estimated sum and the exact sum. This error can be compared to the desired level of accuracy (.01 in this case) to determine if more terms are needed to improve the estimate.