It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils.
Niels Henrik AbelRead
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
Interpretation
What this quote means
The quote questions the validity of applying finite operations to infinite series.
Niels Henrik Abel critiques the common practice of treating infinite series as if they follow the same rules as finite ones. He emphasizes the need for rigorous proof and caution in mathematical operations involving infinite series, suggesting that there are scenarios where such assumptions can lead to incorrect conclusions.
Themes
Infinite SeriesMathematicsDifferentialOperationsValidity
In practice
Example use cases
In a mathematics lecture discussing the convergence of series.
More from Niels Henrik Abel
All quotes →With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
Niels Henrik AbelRead
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