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Mathematics
Problem
For the alternating series
∑
k
=
1
∞
(
−
1
)
k
+
1
1
k
2
\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1}\frac{1}{k^{2}}
k
=
1
∑
∞
(
−
1
)
k
+
1
k
2
1
, let
S
S
S
denote the exact sum and
S
n
S_{n}
S
n
the sum of its first
n
n
n
terms. Explain in one sentence why
∣
S
−
S
n
∣
|S-S_{n}|
∣
S
−
S
n
∣
cannot exceed the magnitude of the
(
n
+
1
)
(n+1)
(
n
+
1
)
st term.
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