Conflict in the answers of the sum of a sequence

Question

We've got the following sequence and we want to calculate its sum: $$D = 2-\frac 43+\frac{10}{9}-\frac{28}{27}+\dots$$
We know that $\lim_{n\to \infty}S_{n}$ would be equal to $t_1/(1-q)$ in geometric equations with $q$ being the ratio and $t_1$ being our first term.

I tried to solve this in two different ways.

A) $$2-(4/3)+(10/9)-(28/27) + ... = (2/3) + (2/27) + (2/243) + ... = 2( (1/3) + (1/27) + (1/243) + ... )$$

$(1/3) + (1/27) + (1/243) + ...$ is a geometric sequence and therefore its sum would be equal to $\frac{t_1}{1-q}$ as the end of our sequence is not apparent.

So the equation mentioned above would be equal to $2 \times (3/8) = (3/4)$

$B)$

$D = 1 + 1 - 1 - (1/3) + 1 + (1/9) - 1 - (1/27) + ...$ $D = (1-1+1-1+...) + (1-(1/3)+(1/9)-(1/27)+...)$
$D = (0$ or $1) + (1-(1/3)+(1/9)-(1/27)+...)$
The second part in $D$ is a geometric equation so its sum would equal $\frac{t_1(1-q^n)}{1-q}$. Therefore $D = (0$ or $1) + (3/4)$ so we can have two answers which are $3/4$ and $7/4$.

Which one(s) is the correct answer to the sum of this sequence? In which have I done anything logically incorrect resulting in these two different answers?

Answer 1

I don't think that here we have a geometric series since the ratio of consecutive terms is not constant. Moreover the series $$\sum_{n=1}^{\infty}\frac{3^{n-1} + 1}{(-3)^{n-1}}=2-\frac 43+\frac{10}{9}-\frac{28}{27} +\frac{82}{81}-\frac{244}{243}+\cdots$$ is not convergent since the generic term does not converge to zero as $n\to +\infty$: $$\left|\frac{3^{n-1} + 1}{(-3)^{n-1}}\right|=1+\frac{ 1}{3^{n-1}}> 1.$$

Answer 2

As Robert Z states, your series is divergent since its terms do not approach $0$. More generally, if the sequence of partial sums does not have a finite limit then the series is divergent. In your series, the partial sums of an even number of terms are approaching $3/4$, while those of an odd number of terms are approaching $7/4$, so the series is divergent. Sometimes divergent series can be assigned a value by specifying a summation method. Your series is, for example, Cesàro summable, with value $5/4$.