$$x = \sum\limits_{n = 0}^\infty {{a^n}} = {1 \over {1 - a}}\,\,\,\,\,\,\,\,\,\,a = 1 - {1 \over x}$$

$$y = \sum\limits_{n = 0}^\infty {{b^n}} = {1 \over {1 - b}}\,\,\,\,\,\,\,\,\,\,b = 1 - {1 \over y}$$

$$z = \sum\limits_{n = 0}^\infty {{c^n}} = {1 \over {1 - c}}\,\,\,\,\,\,\,\,\,\,c = 1 - {1 \over z}$$

$$a,b,c$$ are in $$A.P.$$ OR $$2b = a + c$$

$$2\left( {1 - {1 \over y}} \right) = 1 - {1 \over x} + 1 - {1 \over y}$$

$${2 \over y} = {1 \over x} + {1 \over z} \Rightarrow x,y,z$$ are in $$H.P.$$

We know that

$$e = 1 + {1 \over {1!}} + {1 \over {2!}} + {1 \over {3!}} + ..........$$

and

$${e^{ - 1}} = 1 - {1 \over {1!}} + {1 \over {2!}} - {1 \over {3!}} + .........$$

$$\therefore$$ $$e + {e^{ - 1}} = 2\left[ {1 + {1 \over {2!}} + {1 \over {4!}} + ......} \right]$$

$$\therefore$$ $${1 \over {2!}} + {1 \over {4!}} + {1 \over {6!}} + .......$$

$$ = {{e + {e^{ - 1}}} \over 2} - 1$$

$$ = {{{e^2} + 1 - 2e} \over {2e}}$$

$$ = {{{{\left( {e - 1} \right)}^2}} \over {2e}}$$

If $$n$$ is odd, the required sum is

$${1^2} + {2.2^2} + {3^2} + {2.4^2} + ......$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 2.{\left( {n - 1} \right)^2} + {n^2}$$

$$ = {{\left( {n - 1} \right){{\left( {n - 1 + 1} \right)}^2}} \over 2} + {n^2}$$

[ As $$\left( {n - 1} \right)$$ is even

$$\therefore$$ using given formula for the sum of $$\left( {n - 1} \right)$$ terms.]

$$ = \left( {{{n - 1} \over 2} + 1} \right){n^2} = {{{n^2}\left( {n + 1} \right)} \over 2}$$

$${T_m} = a + \left( {m - 1} \right)d = {1 \over n}...........\left( 1 \right)$$

$${T_n} = a + \left( {n - 1} \right)d = {1 \over m}..........\left( 2 \right)$$

$$\left( 1 \right) - \left( 2 \right) \Rightarrow \left( {m - n} \right)d$$

$$ = {1 \over n} - {1 \over m} \Rightarrow d = {1 \over {mn}}$$

From $$\left( 1 \right)$$ $$a = {1 \over {mn}} \Rightarrow a - d = 0$$