Contents. ??? 1. Introduction ??? 2. Historical Resum??? ??? 3. Averageable Sequences ??? 4. Product Definitions ??? 5. On Certain Possible Definitions Of Summability ??? 6. Definitions Of Evaluability ??? 7. Applications ??? 8. Tests For Cesaro-Summability ??? 9. Theorems On Limits ???10. Conclusion * * * * * An excerpt from: ??? 2. Historical Resum???. The earliest interest in divergent series centers about the series 1 - 1 + 1 - 1 + ... If we assume that this series has a generalized sum s, then the ...
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Contents. ??? 1. Introduction ??? 2. Historical Resum??? ??? 3. Averageable Sequences ??? 4. Product Definitions ??? 5. On Certain Possible Definitions Of Summability ??? 6. Definitions Of Evaluability ??? 7. Applications ??? 8. Tests For Cesaro-Summability ??? 9. Theorems On Limits ???10. Conclusion * * * * * An excerpt from: ??? 2. Historical Resum???. The earliest interest in divergent series centers about the series 1 - 1 + 1 - 1 + ... If we assume that this series has a generalized sum s, then the series, obtained by dropping the first term, - 1 + 1 - 1 + 1... must, by the third fundamental requirement of page 2, also have a generalized sum which is obviously -s. We have then, s - 1 = -s or s = 1/2. Thus, if the series is to have any value at all, that value must be 1/2. And this is precisely the value which Leibniz was led to attach to the series, by different considerations. The sum of n terms of the series is 0 or 1 according as n is even or odd; and since this sum is just as often equal to 1 as it is to 0, its probable value is the arithmetic mean, 1/2. This same value was later attached to the series by Euler, in a more satisfactory, though not entirely rigorous manner. "Let us say that the sum of any infinite series is the finite expression, by the expansion of which the series is generated. In this sense, the sum of the infinite series 1 - x + x 2 - x 3 ... will be 1/(1 + x ), because the series arises from the expansion of the fraction, whatever number is put in place of x." In particular, 1/2 = 1 - 1 + 1 - 1 + ... It is true, as has already been intimated, that none of the methods given above, to prove that the series should have the value 1/2, is satisfactory from a theoretical point of view. But objections have been raised to the result for practical reasons also....
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