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