Does the sum of 2 divergent series diverge?

You may be used to the fact that addition is commutative, it is not the case though when you add up infinitely many numbers. Consider for example the two series and . Both of these series diverge, so if you add them up and then add the results, you get , which is undefined.

Can the sum of a convergent and divergent sequence converge?

If we assume that the sum of the convergent sequence and divergent sequence is convergent, and use that the theorem the book states, both sequences must be convergent. There should be some number that {y_n} converges to but there isn’t, so it can’t be.

Is the product of 2 convergent series convergent?

The product of two absolutely convergent series, where the sum of the absolute values of the terms, is also absolutely convergent. The resulting sum of the product of the two absolutely series can be proven, using the triangle inequality, to be smaller than the product of the limit of each series.

Does the sum of two convergent series converge?

At this point just remember that a sum of convergent series is convergent and multiplying a convergent series by a number will not change its convergence.

Can the sum of two convergent sequences diverge?

The basic idea is that if the terms don’t go to zero, no convergence of the sum, and even if they go to zero but not fast enough, the sum may diverge.

Does the difference of two convergent series converge?

(Note that the above has nothing to do with convergence of any series, since we are adding series it is mathematically sound). Now, note that ∑12n cannot converge, for if it did, say the sum is M, then ∑1n=2M, which is a contradiction as this series doesn’t converge. Hence, the difference does not converge.

Does the sum of convergent series converge?

Do divergent series have a sum?

Divergent series are weird. They certainly don’t have a sum in the traditional sense of the word—that is, their partial sums do not converge (by definition). That said, there are various extensions of the classical notion of “sum” that assign values to divergent sums as well.