Comparison Test/Corollary 1
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Corollary to Comparison Test
Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers.
Let $\sequence {a_n}$ be a sequence in $\R$ or a sequence in $\C$.
Let $H \in \R$.
Let $\exists M: \forall n > M: \size {a_n} \le H b_n$.
Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges.
Proof
Let $\epsilon > 0$.
Then $\dfrac \epsilon H > 0$.
As $\ds \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero.
So:
- $\ds \exists N: \forall n > N: \sum_{k \mathop = n+1}^\infty b_k < \frac \epsilon H$
Let $\sequence {s_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
Then $\forall n > m > \max \set {M, N}$:
\(\ds \size {s_n - s_m}\) | \(=\) | \(\ds \size {\paren {a_1 + a_2 + \cdots + a_n} - \paren {a_1 + a_2 + \cdots + a_m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size {a_{m + 1} + a_{m + 2} + \cdots + a_n}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_{m + 1} } + \size {a_{m + 2} } + \cdots + \size {a_n}\) | Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds H b_{m + 1} + H b_{m + 2} + \cdots + H b_n\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds H \sum_{k \mathop = n + 1}^\infty b_k\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds H \frac \epsilon H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
So $\sequence {s_n}$ is a Cauchy sequence and the result follows from:
or:
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.15$