Math, asked by Anonymous, 2 months ago

Prove that the sequence defined by a1=1 , an = 1 + 1/1! +1/2! + .... 1/(n-1)! n>=2 converges.​

Answers

Answered by Anonymous
1

Answer:

\begin{gathered}{\bf{Given\::}} \\ \end{gathered}

Given:

\begin{gathered}\bf{x = 3\: +\: \sqrt{8}} \\ \end{gathered}

x=3+

8

\begin{gathered} \\ {\bf{To\: Find\::}} \\ \end{gathered}

ToFind:

\begin{gathered} \rm{\left( {x}^{2} \:+ \:\dfrac{1}{ {x}^{2}} \right)} \\ \end{gathered}

(x

2

+

x

2

1

)

\begin{gathered} \\ {\bf{Solution\::}} \\ \end{gathered}

Solution:

First we calculate the value of 'x²'.

➠ \tt{\:x^2\:=\:\left(3\: +\: \sqrt{8}\right)^2}x

2

=(3+

8

)

2

➠ \tt{\:x^2\:=\:9\: +\:8\:+\:6\sqrt{8}\:}x

2

=9+8+6

8

➠ \bf{\:x^2\:=\:17\:+\:6\sqrt{8}\:}x

2

=17+6

8

Now,

We calculate the value of \begin{gathered} \rm{\left(\dfrac{1}{ {x}^{2}} \right)}. \\ \end{gathered}

(

x

2

1

).

➠ \begin{gathered} \tt{\:\dfrac{1}{ {(3\:+\:\sqrt{8})}^{2}} } \\ \end{gathered}

(3+

8

)

2

1

➠ \begin{gathered} \tt{\:\dfrac{1}{9\:+\:8\:+\:6\sqrt{8}} } \\ \end{gathered}

9+8+6

8

1

➠ \begin{gathered} \tt{\:\dfrac{1}{17\:+\:6\sqrt{8}} } \\ \end{gathered}

17+6

8

1

Rationalize the above equation.

➠ \begin{gathered} \tt{\:\dfrac{1}{17\:+\:6\sqrt{8}}\times \dfrac{17\:-\:6\sqrt{8}}{17\:-\:6\sqrt{8}} } \\ \end{gathered}

17+6

8

1

×

17−6

8

17−6

8

➠ \begin{gathered} \tt{\:\dfrac{17\:-\:6\sqrt{8}}{(17\:+\:6\sqrt{8})\:(17\:-\:6\sqrt{8})} } \\ \end{gathered}

(17+6

8

)(17−6

8

)

17−6

8

➠ \begin{gathered} \tt{\:\dfrac{17\:-\:6\sqrt{8}}{(17)^2\:-\:(6\sqrt{8})^2} } \\ \end{gathered}

(17)

2

−(6

8

)

2

17−6

8

➠ \begin{gathered} \tt{\:\dfrac{17\:-\:6\sqrt{8}}{289\:-\:288} } \\ \end{gathered}

289−288

17−6

8

➠ \begin{gathered} \tt{\:\dfrac{17\:-\:6\sqrt{8}}{1} } \\ \end{gathered}

1

17−6

8

➠ \bf{\:(17\:-\:6\sqrt{8})}(17−6

8

)

Now,

Adding these two results that we calculate.

\begin{gathered}:\implies\:\rm{x^2\:+\:\dfrac{1}{ {x}^{2}} } \\ \end{gathered}

:⟹x

2

+

x

2

1

\begin{gathered}:\implies\:\rm{(17\:+\:6\sqrt{8})\:+\:(17\:-\:6\sqrt{8})\:} \\ \end{gathered}

:⟹(17+6

8

)+(17−6

8

)

\begin{gathered}:\implies\:\rm{17\:+\:\cancel{6\sqrt{8}}\:+\:17\:-\:\cancel{6\sqrt{8}}\:} \\ \end{gathered}

:⟹17+

6

8

+17−

6

8

\begin{gathered}:\implies\:\rm{17\:+\:17\:} \\ \end{gathered}

:⟹17+17

\begin{gathered}:\implies\:\bf{34\:} \\ \end{gathered}

:⟹34

⠀⠀⠀\begin{gathered} {\boxed{\bf{\pink{\therefore\:\left( {x}^{2} \:+ \:\dfrac{1}{ {x}^{2}} \right)\:=\:34\:}}}} \\ \end{gathered}

∴(x

2

+

x

2

1

)=34

__________________________

⠀⠀⠀⠀: SOME PROPERTIES :

⠀⠀❶ \bf{(a\:+\:b)^2\:=\:a^2\:+\:b^2\:+\:2ab}(a+b)

2

=a

2

+b

2

+2ab

⠀⠀❷ \bf{(a\:+\:b)\:(a\:-\:b)\:=\:a^2\:-\:b^2\:}(a+b)(a−b)=a

2

−b

2

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