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43 lines
1.2 KiB
Markdown
43 lines
1.2 KiB
Markdown
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#math/calculus
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$\epsilon$=tiniest little value
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N = a possible findable number
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n = index in the sequence.
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L = a possible findable limit
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# convergence (for sequences)
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for all $\epsilon>0$ we can find $N$:
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$n>N$ ------- $|L-a_n|<\epsilon$
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# divergence (for sequences)
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for all $M$ we can find $N$ such that (st):
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$n>N$ ------ $a_n>M$
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The sequence constantly grows in a direction.
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Or, it can oscillate! $sin(x)$ does not converge, so it's divergent.
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# test for divergence or convergence of series
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```python
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from sympy import *
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x=symbols('x')
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series = Sum(1/(6 + exp(-x)), (x, 1, oo))
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series.is_convergent()
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```
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- test for divergence
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- break it into parts
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- check if geometric
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- telescoping
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- MCT [[monotone_and_bounded]]? Integral bigger converges? Series converges
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- Integral test. If the integral starting from 1 converges, it converges. For the Integral test, the function must be:
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- continuous
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- positive
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- decreasing
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![[Pasted image 20220524104017.png]]
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- ![[More tests for convergence#Direct Comparison Test]]
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- ![[More tests for convergence#Limit Comparison Test]]
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- ![[Alternating Series Test]]
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- ![[Ratio test]]
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- Root Test ![[Pasted image 20220526191532.png|Root Test]]
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