Then, as is mentioned above, these functions can be expanded by uniformly and absolutely converging in the D̅ Fourier series, f_{i}(r,varphi =frac{1}{2}a_{0}^{(i)}(r)+ sum_{m=1}^{infty } bigl( a_{m}^{(i)}(r) cos mvarphi+b_{m}^{(i)}(r)sin mvarphi bigr), (32) where left.
Moreover, for (p = q + 1) it absolutely converges on (left| z right| = 1) if the condition A^ = text{Re} left( {sumlimits_{j = 1}^{q} {b_{j} } - sumlimits_{j = 1}^{q + 1} {a_{j} } } right) > 0,holds and is conditionally convergent for (left| z right| = 1) and (z ne 1) if (- 1 < A^ le 0) and is finally divergent for (left| z right| = 1) and (z ne 1) if (A^ le - 1).
end{aligned} (3.12) Since (sum_{n=1}^{infty}e_{n}) converges absolutely, (3.11) converges absolutely.
This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that :\left\|\mathbf{L}-\sum_{k=0}^N\mathbf{x}_k\right\|\to 0\quad\text{as }N\to\infty.
end{aligned} (3.10) Because (sum_{n=1}^{infty}e_{n}) converges absolutely, (3.10) is sufficient to establish that (3.9) is absolutely convergent.
With ρ ∞ < 1, the second sum in (26) converges absolutely for all constants B, C, E > 0. Therefore, the series in (26) is absolutely convergent for sufficiently-high k.
Completeness of the space holds provided that whenever a series of elements from ℓ2 converges absolutely (in norm), then it converges to an element of ℓ2.
provided the sums converge absolutely.
The modified zeta functions ∑n∈Kn−s, where K⊂N, converge absolutely for Res>1.
