is absolutely convergent and no matter how we reorganize the terms of this series, we always get the same value. In fact, we can show that the value of this series is (n^2+n > 5) for (n >1text{,}) and therefore this series diverges because the harmonic series diverges. So when the original series converges, it does so conditionally. A typical conditionally convergent integral is that on the non-negative real axis of sin ( x 2 ) {textstyle sin(x^{2})} (see Fresnel integral). By definition, a series converges conditionally when it converges but diverges. Conversely, it is questionable whether it is possible to converge as long as it diverges. The following sentence shows that this is not possible. The fundamental question we want to answer about a series is whether the series converges or not. If a series has both positive and negative terms, we can refine this question and ask ourselves whether or not the series converges when all the terms are replaced by their absolute values. It is the distinction between absolute and conditional convergence that we examine in this section. Now that we assume that (sum {left| {{a_n}} right|} ) is convergent, and then (sum {2left| {{a_n}} right|} ) is also convergent, because one can simply factor the 2 of the series and 2 times a finite value is always finite. However, this allows us to use the comparison test to say that (sum left({{a_n} + left| {{a_n}} right|} right) ) is also a convergent series. A series (displaystyle sum {{a_n}} ) is said to be absolutely convergent if (displaystyle sum {left| {{a_n}} right|} ) is convergent.

If (displaystyle sum {{a_n}} ) is convergent and (displaystyle sum {left| {{a_n}} right|} ) is divergent is called the conditionally convergent series. Thus, the series (dssum_{n=0}^infty (-1)^{n}{3n+4over 2n^2+3n+5}) converges through the alternating series test, and we conclude that the series converges conditionally. Thus, since the series of absolute values is divergent, the original series in the question is not absolutely convergent. Even if this series is not absolutely convergent, it can still be conditionally convergent. That is what we will check next. So, I`m just going to create a space to do it. To check the conditional convergence, we check whether the original series is convergent or not. One thing to keep in mind about this series is that the negative power from one to n two creates a changing effect between positive and negative values. This allows us to use the alternating series test. This fact is one of the ways in which absolute convergence is a “stronger” type of convergence. Lines that are absolutely convergent are guaranteed to be convergent. However, convergent series may or may not be absolutely convergent.

Bernhard Riemann proved that a conditionally convergent series can be rearranged in such a way that it converges to any value, including ∞ or −∞; see Riemann series theorem. The Levy–Steinitz theorem identifies the set of values to which a series of terms in Rn can converge. and so (sum {{a_n}} ) is the difference between two convergent and therefore also convergent series. If (sum a_n) converges, but not (sum |a_n|), we say that (sum a_n) converges conditionally. Recall that the alternating harmonic series (dssum_{n=1}^{infty} frac{(-1)^{n-1}}{n}) converges, but the corresponding set of absolute values, namely the harmonic series (dssum_{n=1}^{infty}frac{1}{n}text{,}) diverges. Therefore, the alternating harmonic series is conditionally convergent. and therefore the series diverges (since the harmonic series is divergent). Therefore, the original series diverges or is conditionally convergent. We apply the alternating series test: Therefore, the original series is absolutely convergent (and therefore convergent). Determine whether each row is absolutely convergent, conditionally converged, or divergent.

First of all, as we showed above in Example 1a, an alternating harmonic is conditionally convergent, and regardless of the value we have chosen, there is a rearrangement of the terms that give this value. Also note that this fact does not tell us what this rearrangement should be, only that it exists. The Riemann series theorem states that by an appropriate rearrangement of terms, a conditionally convergent series can converge or diverge to any desired value. The Riemann series theorem can be proved by first taking just enough positive terms to exceed the desired limit, then taking just enough negative terms to fall below the desired limit, and repeating this method. Since the terms of the original series tend towards zero, the rearranged series converges towards the desired limit. A slight variation causes the new series to diverge into positive infinity or negative infinity. In other words, if a series converges, then the series must also converge. It`s not hard to understand why this is true. .