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limit_in_a_metric_space [2015/11/26 22:28] nikolaj |
limit_in_a_metric_space [2017/02/02 21:25] nikolaj |
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| === Examples === | === Examples === | ||
| - | $\sum_{k=1}^\infty a_n = y$ | + | $\sum_{k=1}^\infty a_k = y$ |
| means | means | ||
| - | $ \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n a_n - y \, |<\varepsilon $ | + | $ \forall (\varepsilon\in{\mathbb R}_{>0}).\,\exists (m\in{\mathbb N}).\,\forall (n\ge_{\mathbb N} m).\,| \sum_{k=1}^n a_k - y \, |<\varepsilon $ |
| == Infinite sum == | == Infinite sum == | ||
| Line 57: | Line 57: | ||
| $\int_a^\infty f(x)\,{\mathrm d}x := \lim_{b\to \infty} \int_a^b f(x)\,{\mathrm d}x$ | $\int_a^\infty f(x)\,{\mathrm d}x := \lim_{b\to \infty} \int_a^b f(x)\,{\mathrm d}x$ | ||
| + | |||
| + | === sci === | ||
| + | |||
| + | >>7944543 | ||
| + | I take one for the notebook: | ||
| + | |||
| + | In calculus/analysis, infinity isn't used as an entity (like a number), but instead | ||
| + | >limit n to infinity | ||
| + | means | ||
| + | >for whatever m you choose (arbitry), there is such and such, that such and such | ||
| + | |||
| + | For example you may consider a sequence [math] s_n [/math] given by | ||
| + | 1/2, 1/4, 1/8, 1/16, ... | ||
| + | then the "limit of n to infinity" is the number y=0. | ||
| + | Why? Because for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that [math] s_n [/math] became smaller than [math] \varepsilon [/math]. | ||
| + | For example, choose the small number [math] \varepsilon = 0.0003 [/math]. The sequence actualy becomes forever smaller than that after, say, m = 4000. Indeed, for any n after 4000, the number [math] s_n [/math] is something smaller than 1/4000, which is 0.00025. | ||
| + | |||
| + | So the limit to infinity is formalized as something to do with | ||
| + | >for arbitrary high values of the index, the thing itself is still restricted. | ||
| + | |||
| + | Coming back to the definition: | ||
| + | The limit of n to infinity of a seqeunce [math] s_n [/math] is y, if for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that the difference (here given by the distance on the real number line) between the value [math] s_n [/math] and this y became smaller than [math] \varepsilon [/math]. | ||
| + | |||
| + | In formulas | ||
| + | |||
| + | [math] \lim_{n \to \infty} s_n = y[/math] | ||
| + | |||
| + | iff | ||
| + | |||
| + | [math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) . \, \exists ( m \in {\mathbb N} ) . \, \forall ( n \ge_{\mathbb N} m) . \, | s_n - y \, |<\varepsilon [/math] | ||
| + | |||
| + | Another example: Consider again the sequence [math] s_n [/math] given by | ||
| + | 1/2, 1/4, 1/8, 1/16, ... | ||
| + | and create a new sequence | ||
| + | |||
| + | [math] S_m = \sum_{k=1}^n a_n [/math] | ||
| + | |||
| + | which has members | ||
| + | 1/2, 1/2+1/4, 1/2+1/4+1/8, ... | ||
| + | |||
| + | You can prove that with y=1 the above formula regarding [math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) [/math] holds. | ||
| + | |||
| + | So we say | ||
| + | |||
| + | [math] \sum_{k=0}^\infty s_n := \lim_{n \to \infty} \sum_{k=0}^n s_k = 1 [/math] | ||
| === Reference === | === Reference === | ||
