Precision of numerical series value in Wolfram|Alpha

Questions
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1

Error

It is gives

\sum_{k=0}^{\infty}\frac{(-1)^k}{k^2+2k+2}\approx 0.363998

but exact value:

\sum_{k=0}^{\infty}\frac{(-1)^k}{k^2+2k+2}=\frac{1}{2}-\frac{\pi}{e^{\pi}-e^{-\pi}}\approx 0.3639854725089334
2

With Mathematica 13.1,numerically:

 NSum[(-1)^k/(k^2 + 2 k + 2), {k, 0, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 32]
(*0.36398547250893341852488170865923*)

NSum[(-1)^k/(k^2 + 2 k + 2), {k, 0, Infinity}, 
 Method -> "WynnEpsilon", NSumTerms -> 10^5, WorkingPrecision -> 32]
(*0.36398547250893341852488170816398*)

Symbolically:

Sum[(-1)^k/(k^2 + 2 k + 2), {k, 0, Infinity}]
(*1/2 (1 - \[Pi] Csch[\[Pi]])*)
N[%, 32]
(*0.36398547250893341852488170865923*)

If you use WolframAlfa answer is not precisie ,because for quick answer to you from web.
Try to put:

N[Sum[(-1)^k/(k^2 + 2 k + 2), {k, 0, Infinity}] ,32]

Give you:
0.36398547250893341852488170816398

3

Thanks.
Yes, I used Wolframa|Alpha.
But I think that it is not a good idea to have more velocity and error results from web form.