Answer:
Proofs begin with one or more given statements, which are provided. ... In a two column proof, the statements are written in one column, and the reasons are written next to them in a second column. A flow proof uses a diagram of to show each statement leading to the conclusion
Step-by-step explanation:
Answer:
Given series is divergent
Step-by-step explanation:
<u><em>Step(i):-</em></u>
By using Ratio test
![\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = l](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7Ba_%7Bn%2B1%7D%20%7D%7Ba_%7Bn%7D%20%7D%20%20%7C%20%3D%20l)
a)
'l' is finite then the given ∑aₙ is convergent
b)
Here 'l' is infinite then the ∑aₙ is divergent
<u><em>Step(ii):-</em></u>
Given aₙ = ![\frac{n!}{n}](https://tex.z-dn.net/?f=%5Cfrac%7Bn%21%7D%7Bn%7D)
![a_{n+1} = \frac{(n+1)!}{n+1}](https://tex.z-dn.net/?f=a_%7Bn%2B1%7D%20%3D%20%5Cfrac%7B%28n%2B1%29%21%7D%7Bn%2B1%7D)
![\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} |\frac{\frac{(n+1)!}{n+1} }{\frac{n!}{n} } |](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7Ba_%7Bn%2B1%7D%20%7D%7Ba_%7Bn%7D%20%7D%20%20%7C%20%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7B%5Cfrac%7B%28n%2B1%29%21%7D%7Bn%2B1%7D%20%7D%7B%5Cfrac%7Bn%21%7D%7Bn%7D%20%7D%20%7C)
we know that n ! = n (n-1) (n-2) ......3.2.1
and also (n+1) ! = (n+1)n!
![\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} |\frac{\frac{n+1)n!}{n+1} }{\frac{n!}{n} } |](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7Ba_%7Bn%2B1%7D%20%7D%7Ba_%7Bn%7D%20%7D%20%20%7C%20%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7B%5Cfrac%7Bn%2B1%29n%21%7D%7Bn%2B1%7D%20%7D%7B%5Cfrac%7Bn%21%7D%7Bn%7D%20%7D%20%7C)
![\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} n](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7Ba_%7Bn%2B1%7D%20%7D%7Ba_%7Bn%7D%20%7D%20%20%7C%20%3D%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20n)
= ∞
<em>Given sum of the series is divergent</em>
Answer:
0.036
Step-by-step explanation:
0.4 × 0.09 = 0.036
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Answer:
or
Step-by-step explanation:
we know that
Applying Pythagoras' Theorem, calculate how many kilometers eastward Santos walked
so
Let
x----> quantity of kilometers eastward that Santos walked
![x^{2}=5^{2}-2^{2}](https://tex.z-dn.net/?f=x%5E%7B2%7D%3D5%5E%7B2%7D-2%5E%7B2%7D)
![x^{2}=21](https://tex.z-dn.net/?f=x%5E%7B2%7D%3D21)
----> exact value
-----> approximate value
Answer:
<em>It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics.</em>
Step-by-step explanation:
Hope It Help