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.

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

$$ \sum_{j=0}^q (-1)^{q-j} \beta_j. $$