Question on Morse inequalities I want to understand why: From K.C Chang's book "Infinite Dimensional
Morse Theory and Multiple Solution Problems":
if i have 

then $(4.1)$ is formal : it means that 

EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$
Whene $t=-1$ we have directly that $\displaystyle\sum_{q=0}^{\infty}(-1)^q M_q=\displaystyle\sum_{q=0}^{\infty} (-1)^q \beta_q$
and wehave also that $\displaystyle\sum_{q=0}^{\infty}(-1)^q M_q\geq\displaystyle\sum_{q=0}^{\infty} (-1)^q \beta_q$ because $Q(t)$ has nonnegative coefficient 
but how to obtain that $\displaystyle\sum_{j=0}^{q}(-1)^{q-j} M_j\geq\displaystyle\sum_{j=0}^{q} (-1)^{q-j} \beta_j$ ???
I ask this question on Mathematics stackex change but they don't answer me and they told me to ask it here 
i need your help please help me
thank you
 A: Let us set
$$ M(t)=\sum_{q=0}^\infty M_tt^q,\;\;P(t)=\sum_{q=0}^\infty \beta_q t^q. $$
We can rewrite  (4.1) as 
$$ M(t)=P(t)+(1+t)Q(t),  $$
where the formal power series $Q(t)$ has nonnegative coefficients. We deduce
$$ (1+t)^{-1} M(t)=(1+t)^{-1}P(t) +Q(t). \tag{1}$$
This shows that the Taylor coefficients of  $(1+t)^{-1} M(t)$ are    $\geq $ than the corresponding Taylor coefficients of $(1+t)^{-1}P(t)$. To compute these Taylor  coefficients use the know expansion
$$(1+t)^{-1}=\sum_{q=0}^\infty(-1)^q t^q. $$
Using this in (1) you obtain all the  Morse inequalities.
The coefficient of $t^q$ in $(1+t)^{-1}M(t)$ is
$$ \sum_{\substack{k+j=q\\k,j\geq 0}}(-1)^k M_j\stackrel{k=q-j}{=}\sum_{j=0}^q (-1)^{q-j} M_j $$.
This coefficient is $\geq $ than the coefficient of $t^q$ in $(1+t)^{-1}P(t)$ which  is
$$ \sum_{j=0}^q (-1)^{q-j} \beta_j. $$
A: i have search google and i find a link where you can download this book :
http://www.doc88.com/p-607135346844.html
i read your problem in this book roughly (theorem 4.3 page 37 )
i also find the solution to your question in this book , it is included in the proof of theorem 4.3 itself actually !
you should read its proof carefully, your question is a consequence of Corollary 4.1 and Theorem 4.2 , the answer is in its proof indeed !
