![]() ![]() One type of series that plays a major role in our study is that of a convergent -series,įor. As such, we should we care to use the sharper estimate if we Terms versus terms is desirable, a computer can perform both of these calculationsĪlmost instantaneously. ![]() While it seems that the second result is “nicer” in the sense that adding We hope that the reader actually performed the computations in the previousĮxample. Of terms needed to approximate the value of a series. This in practice will greatly reduce the number In going from the penultimate to the last step, we are justified in our calculationīecause the integrals both converge. The infinite series must be within of its minimum possible value (the lefthand side)Īnd its maximum possible value (the righthand side). If the difference between the left and right sides of the inequality is less than, then Upper estimate for the series and the lower estimate be no more than. Series to within of its actual value, we can require that the difference between the This leads to an important observation if we want to approximate a convergent Since we have a positivity assumption on the terms, note that this means that forĮvery, will be an overestimate underestimate.
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