Phone turns off when battery is about 40%

Hi, I have installed Android 6.0 few weeks ago. Since last week I have noticed some weird behaviour.
While discharging, when the battery is about 40%, my Nexus 5 turns off immediately. Then, when I am trying to turn it on, I got a low battery warning and can't turn it on. When I connect it to the charger and turn it on, the battery indicator shows about 15-20%.
Is my battery going to die? It looks normal, not swollen. I tried reinstalling Marshmallow (with wipe) but it didn't help.

When reinstalling system, I noticed an other problem. Stock recovery doesn't work. After flashing it with

Code:

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fastboot erase recovery
fastboot flash recovery recovery.img

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I have only the laying android and red triangle picture. TWRP works when flashed the same way. But it disappears after reboot!

What's wrong with my Nexus? Any ideas? What can I try to do?

HELP! Note 4 Turns On No Display!

Hi, my att note 4 dropped from a small hill on to the sidewalk in about 20 feet in the air with a case, the case was made of a hard gel typish case and the case twists when you put twist the case . After that I picked up the note 4, there were no cracks and then when i turned the phone on I saw a led light when the Samsung boot animation comes but there was no display. I took out the spen and i started tapping on my pin number on my lock screen and i heard vibrations when putting the password. Any ideas about what to do? Please help im a student and this is the phone i have:crying: