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Infinite  Interval

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☼ Infinite  Interval :–

  1. ( a , ∞ )  =   { x /    x  >  a,  x ∈ R }       

2. [ a , ∞ ) =   { x /    x  ≥  a,  x ∈ R }

  1. ( – ∞ , b )  =   { x /    x  <  b,  x ∈ R }                                                                             

                                                                                           

  1. ( – ∞ , b ]  =   { x /    x  ≤  b,  x ∈ R }            

☼ log x  =  loge x    i.e.  If  the  base  of  logarithm  is  not  mentioned  then  it  is  taken  by  e .

In  Higher  Studies ,

ln  x   =   loge x            &          log x  =  log10 x

☼ logb a    =    \frac{\mathrm{lo}\mathrm{g}_ca}{\mathrm{lo}\mathrm{g}_cb}        ;        Where  c \in R^+ –  { 1 }

☼ logb a    =    \frac{1}{\mathrm{lo}\mathrm{g}_ab}   ;        Where  a, b \in R^+ –  { 1 }   ,  a^{\log{x}} =x^{log\ a}

☼ logx x   =   1  ,        logx 1   =   0  ,    logb am  =   m  logb a  ,
b^{\log_b\ a}=\ a

☼ \sum_{i=1}^{n}1  =  \sum1  =  1  +  1  +  1  +  . . .  +  1 (n times)   =   n

☼ \sum_{i=1}^{n}i = \sum n  =  1  +  2  +  3  + . . . +  n   = \frac{n(n+1)}{2}

☼ \sum_{i=1}^{n}i^2 = \sum n^2  =    1^2 + 2^2 + 3^2 + . . . + n^2  =  \frac{n(n+1)(2n+1)}{6}

☼ \sum_{i=1}^{n}i^3 = \sum n^3 = 1^3 + 2^3 + 3^3 + . . . + n^3  =  \frac{n^2(n+1)^2}{4} = \left[\sum n\right]^2

☼ \sum{(2n-1)} = 1 + 3 + 5 + . . . + (2n - 1) = n^2

 

☼ n Pr =  \frac{n^!}{(n-r)!}

=   n  ( n – 1 ) ( n – 2 ) . . . ( n – r  + 1 )

☼ \left(\begin{matrix}n\\r\\\end{matrix}\right)=  n C r  =  \frac{n^!}{(n-r) !^r!}  = \frac{_nP_r}{r!}

☼ \left(\begin{matrix}n\\0\\\end{matrix}\right) = \left(\begin{matrix}n\\n\\\end{matrix}\right)  =  1,   \left(\begin{matrix}n\\1\\\end{matrix}\right)  =  \left(\begin{matrix}n\\n-1\\\end{matrix}\right) = n      &  \left(\begin{matrix}n\\r\\\end{matrix}\right) = \left(\begin{matrix}n\\n-r\\\end{matrix}\right)

☼ (x+y)^n = \left(\begin{matrix}n\\0\\\end{matrix}\right)x^n +  \left(\begin{matrix}n\\1\\\end{matrix}\right)x^{n-1}y  +  \left(\begin{matrix}n\\2\\\end{matrix}\right)x^{n-2}y^2 +  \left(\begin{matrix}n\\3\\\end{matrix}\right)x^{n-3}y^3 +    . . .  +  \left(\begin{matrix}n\\n\\\end{matrix}\right)y^n

☼ (1+x)^n = \left(\begin{matrix}n\\0\\\end{matrix}\right) +  \left(\begin{matrix}n\\1\\\end{matrix}\right)x  +  \left(\begin{matrix}n\\2\\\end{matrix}\right)x^2  +  \left(\begin{matrix}n\\3\\\end{matrix}\right)x^3 +    . . .  +  \left(\begin{matrix}n\\n\\\end{matrix}\right)x^n

☼ \left(\begin{matrix}n\\0\\\end{matrix}\right) + \left(\begin{matrix}n\\1\\\end{matrix}\right) +  \left(\begin{matrix}n\\2\\\end{matrix}\right) + \left(\begin{matrix}n\\3\\\end{matrix}\right) +    . . .  +  \left(\begin{matrix}n\\n\\\end{matrix}\right) = 2^n

☼ In the expansion of  (x+y)^n,    T_{r+1} = \left(\begin{matrix}n\\r\\\end{matrix}\right) x^{n-r}yr

☼ In Arithmetic Progression,

T_n=  a  +  (n – 1) d                    ;   a = First Term

S_n= \frac{n}{2} [  2a +  (n – 1) d  ]       ;   d = Common difference

=  \frac{n}{2} ( a + l )                        ;    l = T_n

☼ In Geometric Progression,

T_n= a r^{n-1}      ;  a =  First Term  &  r  =  Common Ratio   

S_n= \frac{a(r^n-1)}{r-1}     ;     r  > 1

=  \frac{a(1-r^n)}{1-r}     ;          r  < 1

 =   n a                 ;          r  =  1

S_n= \frac{a-r^T_n}{1-r} ; \ \ r \neq 1

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